Algebraic Expressions
Algebra · 102 lessons
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An **algebraic expression** is an expression involving letters and/or numbers called factors multiplied together. ### Addition and Subtraction of Algebraic Expressions quote **Example:** $$(3x^2 - 7ab + 2e)$$ This has three terms. The coefficients are: $$3,\ -7,\ and\ 2$$ For the second term, $-7ab$, the factors are: $$-7,\ a,\ and\ b$$ **NOTE:** We can only add or subtract like terms (terms that contain the same variables raised to the same power). **Example 1:** $$Simplify -2[-3(x - 2y) + 4y]$$ **Step 1:** Distribute $-3$ inside the first brackets $(x - 2y)$ $$-3(x - 2y) = -3(x) + (-3)(-2y)$$ $$-3(x - 2y) = -3x + 6y$$ **Step 2:** Rewrite the expression inside the brackets $$-2[-3(x - 2y) + 4y]$$ $$becomes:$$ $$-2[-3x + 6y + 4y]$$ **Step 3:** Combine like terms inside the brackets $$-3x + 6y + 4y = -3x + 10y$$ **Step 4:** Rewrite the expression $$-2[-3x + 6y + 4y]$$ $$become:$$ $$-2x[-3x+10y]$$ **Step 5:** Distribute $-2$ to each term inside the brackets $$-2[-3x + 10y] = (-2)(-3x) + (-2)(10y)$$ $$-2[-3x + 10y] = 6x - 20y$$ **Final Answer:** $6x - 20y$ **Example 2:** $$Simplify: 5x[-4 + 10(x - y)] + 7x$$ **Step 1:** Distribute $10$ inside the brackets $(x - y)$ $$10(x - y) = 10(x) + 10(-y)$$ $$10(x - y) = 10x - 10y$$ **Step 2:** Rewrite the expression inside the brackets $$5x[-4+10(x-y)]+7x$$ $$becomes:$$ $$5x[-4 + 10x - 10y] + 7x$$ **Step 3:** Distribute $5x$ to each term inside the brackets $$5x[-4x+10x-10y] = 5x(-4) + 5x(10x) + 5x(-10y)$$ $$5x[-4+10x-10y] = -20x + 50x^2 - 50xy$$ **Step 4:** Rewrite the complete expression $$5x[-4+10x-10y]+7x$$ $$becomes:$$ $$-20x + 50x^2 - 50xy + 7x$$ **Step 5:** Combine like terms $$Combine the x terms: -20x + 7x = -13x$$ $$Keep other terms as they are$$ **Final Answer:** $50x^2 - 13x - 50xy$ quote ### Multiplication of Algebraic Expressions quote $$(a + b)(x + y) = ax + ay + bx + by$$ Algebraic expressions with exponents or powers can be combined **if the bases are the same**. **Example:** $$2x^3 + 5x^3 = 7x^3$$ But: $$2x^3 + 5x$$ cannot be combined because the bases $x^3$ and $x$ are not the same. **Example 1:** $$Multiply (2x + 3)(x^2 - x - 5)$$ **Step 1:** Distribute the first term $2x$ to each term in the second polynomial $$2x(x^2 - x - 5) = 2x(x^2) + 2x(-x) + 2x(-5)$$ $$2x(x^2 - x - 5) = 2x^3 - 2x^2 - 10x $$ **Step 2:** Distribute the second term $3$ to each term in the second polynomial $$3(x^2 - x - 5) = 3(x^2) + 3(-x) + 3(-5)$$ $$3(x^2 - x - 5) = 3x^2 - 3x - 15 $$ **Step 3:** Rewrite the complete expression $$(2x + 3)(x^2 - x - 5)$$ $$becomes:$$ $$2x^3 - 2x^2 - 10x + 3x^2 - 3x - 15$$ **Step 4:** Group like terms together $$2x^3 + (-2x^2 + 3x^2) + (-10x - 3x) - 15$$ **Step 5:** Combine like terms $$x^2 terms: -2x^2 + 3x^2 = x^2$$ $$x terms: -10x - 3x = -13x$$ $$Constant term: -15$$ **Final Answer:** $2x^3 + x^2 - 13x - 15$ **Example 2:** $$Expand (x - 3)^2$$ **Step 1:** Rewrite the expression as multiplication $$(x - 3)^2 = (x - 3)(x - 3)$$ **Step 2:** Distribute the first term $x$ to each term in the second polynomial $$(x - 3) = x(x) + x(-3)$$ $$x(x - 3) = x^2 - 3x$$ **Step 3:** Distribute the second term $-3$ to each term in the second polynomial $$ -3(x - 3) = -3(x) + (-3)(-3)$$ $$-3(x - 3) = -3x + 9 $$ **Step 4:** Rewrite the complete expression $$(x - 3)(x - 3)$$ $$becomes:$$ $$x^2 - 3x - 3x + 9$$ **Step 5:** Combine like terms $$x terms: -3x - 3x = -6x$$ $$Keep other terms as they are$$ **Final Answer:** $x^2 - 6x + 9$ **Example 3:** $$Expand (a - 2y^2)^2$$ **Step 1:** Rewrite the expression as multiplication $$(a - 2y^2)^2 = (a - 2y^2)(a - 2y^2)$$ **Step 2:** Distribute the first term $a$ to each term in the second polynomial $$a(a - 2y^2) = a(a) + a(-2y^2)$$ $$a(a - 2y^2) = a^2 - 2ay^2 $$ **Step 3:** Distribute the second term $-2y^2$ to each term in the second polynomial $$-2y^2(a - 2y^2) = -2y^2(a) + (-2y^2)(-2y^2)$$ $$-2y^2(a - 2y^2) = -2ay^2 + 4y^4 $$ **Step 4:** Rewrite the complete expression $$(a - 2y^2)(a - 2y^2)$$ $$becomes:$$ $$a^2 - 2ay^2 - 2ay^2 + 4y^4$$ **Step 5:** Combine like terms $$ay^2 terms: -2ay^2 - 2ay^2 = -4ay^2$$ $$Keep other terms as they are$$ **Final Answer:** $a^2 - 4ay^2 + 4y^4$ quote ## Division of Algebraic Expressions quote Algebraic expressions with exponents or powers can be combined **if the bases are the same**. **Example 1:** $$3x^7{x^2}$$ **Step 1:** Identify the expression $$We have: 3x^7{x^2}$$ $$This is a fraction with like bases (x) in numerator and denominator.$$ **Step 2:** Apply the quotient rule for exponents $$3x^7{x^2} = 3x^{7-2}$$ **Step 3:** Write the final simplified form $$3x^{(7-2)}$$ $$becomes:$$ $$3x^5$$ **Final Answer:** $3x^5$ **Example 2:** $$-6a^2bc^2{2abc^4}$$ **Step 1:** Separate the coefficients from the variable terms $$-6a^2bc^2{2abc^4} = (-6{2}) ( a^2bc^2{abc^4} )$$ **Step 2:** Simplify the coefficient $$= (-3) ( a^{2-1}b^{1-1}c^{2-4} )$$ $$= (-3) ( a^1 b^0 c^{-2} )$$ $$= (-3) ( a{c^2} )$$ **Step 3:** Write the final simplified form $$(-3) ( a{c^2} )$$ $$becomes:$$ $$-3a{c^2}$$ **Final Answer:** $-3a{c^2}$ **Example 3:** $$(x + 4)(x - 3){(x - 3)}$$ **Step 1:** Identify the common factor The common factor is $(x - 3)$ in both numerator and denominator. **Step 2:** Cancel the common factor $$(x + 4)(x - 3){(x - 3)} = (x + 4) 1$$ $$The (x - 3) terms cancel out.$$ **Step 3:** Write the simplified expression $$(x + 4) 1 = x + 4$$ **Final Answer:** $x + 4$ quote
Algebraic Expressions
Simplify: $ -2[3x - 2(4x - 5y)] + 6y $
Why A is correct:
Work inside the brackets first: 4x - 5y stays as is, then 2(4x - 5y) = 8x - 10y. So 3x - (8x - 10y) = 3x - 8x + 10y = -5x + 10y. Then -2(-5x + 10y) = 10x - 20y. Finally, 10x - 20y + 6y = 10x - 14y. ✓
Why others are wrong:
- B (-2x + 6y): Made an error in the distribution step—likely didn't properly distribute the negative sign through the brackets.
- C (-6x + 10y): Forgot to apply the outer -2 multiplier; only simplified what was inside the brackets.
- D (2x - 4y): Combined distribution errors or lost track of signs during simplification.
Expand and simplify: $ (x - 2)(x^2 + 2x + 4) $
Why A is correct:
This is the difference of cubes formula: (a - b)(a² + ab + b²) = a³ - b³. Here a = x and b = 2, so (x - 2)(x² + 2x + 4) = x³ - 2³ = x³ - 8.
Why the others are wrong:
- B: Results from incorrectly multiplying or combining terms; the x² and constant terms shouldn't appear in the final answer.
- C: This looks like someone tried to expand but made errors—it has too many terms and the wrong coefficients.
- D: Missing the expansion of the x² · x term entirely, leaving out major parts of the product.
Factor completely: $ 9x^2 - 24xy + 16y^2 $
Why A is correct:
This is a perfect square trinomial. Check: (3x - 4y)² = (3x)² - 2(3x)(4y) + (4y)² = 9x² - 24xy + 16y². The middle term -24xy matches perfectly.
Why the others are wrong:
- B: (9x - 16y)² gives 81x² - 288xy + 256y², which doesn't match our original expression.
- C: This isn't even factored form (it's just simplified differently), and 3x² - 4y² doesn't equal our original trinomial.
- D: (3x + 4y)² gives 9x² + 24xy + 16y² with a *positive* middle term, but we need negative.
Simplify: $ \frac{6a^3b^2 - 3a^2b^3}{3a^2b^2} $
Correct Answer: A (2a - b)
Split the fraction into two parts: (6a³b²)/(3a²b²) − (3a²b³)/(3a²b²)
- First part: 6a³b² ÷ 3a²b² = 2a (divide coefficients and subtract exponents)
- Second part: 3a²b³ ÷ 3a²b² = b (same process)
- Result: 2a − b ✓
Why others are wrong:
- B (2a + b): Sign error—you subtract the second term, not add it
- C (2a² − b²): You incorrectly kept exponents instead of canceling them out
- D (3a − b): Wrong coefficient on *a*—should divide 6 by 3, not keep it as 3
What is the simplified form of $ (x + 2)^2 - (x - 2)^2 $?
Correct answer: 8x
Expand both squares: (x + 2)² = x² + 4x + 4 and (x - 2)² = x² - 4x + 4. Subtracting them gives (x² + 4x + 4) - (x² - 4x + 4) = x² + 4x + 4 - x² + 4x - 4 = 8x.
Why the others are wrong:
- B (4x): You'd get this if you only subtracted the middle terms (4x - (-4x) = 8x), but forgot to distribute the negative sign properly through the second expression.
- C (4): This forgets that the x² terms cancel out—the answer must still include x.
- D (2x + 4): This comes from incorrectly expanding or combining terms; the constant terms actually cancel.
Simplify: $ 3x[2x - (4x - 5)] - 6 $
Why A is correct:
Work from the inside out: 2x - (4x - 5) = 2x - 4x + 5 = -2x + 5. Then 3x(-2x + 5) = -6x² + 15x. Finally, -6x² + 15x - 6 is your answer.
Why the others are wrong:
- B (6x^2 - 12x - 6): This has the wrong sign on x²; you'd get this if you made a sign error with the negative in front of the parentheses.
- C (-2x^2 + 5x - 6): This forgot to multiply -2x + 5 by 3x; it only simplified the brackets.
- D (-6x^2 + 9x - 6): This made an arithmetic error when multiplying 3x by 5 (got 9x instead of 15x).
Factor: $ x^2 + 6x + 9 - (x + 3)^2 $
Why A is correct:
The expression x² + 6x + 9 is a perfect square trinomial that factors to (x + 3)². So when you subtract (x + 3)² from it, you get (x + 3)² − (x + 3)² = 0.
Why the others are wrong:
- B (6x): This would be part of the expanded form, but doesn't account for the subtraction of the entire squared term.
- C (x² + 6x): This is just the first two terms of the original trinomial; it ignores both the 9 and the subtraction.
- D (x² + 9): This cherry-picks terms but misses the 6x term and doesn't recognize that the whole expression cancels out.
Expand and simplify: $ (2x - 3)^2 - (x + 1)^2 $
Why A is correct:
Expand (2x - 3)² = 4x² - 12x + 9 and (x + 1)² = x² + 2x + 1, then subtract: (4x² - 12x + 9) - (x² + 2x + 1) = 3x² - 14x + 8. ✓
Why others are wrong:
- B & D: These have x² as the leading term, but subtracting x² from 4x² gives 3x², not 1x².
- C: This results from an arithmetic error when combining like terms (likely miscalculating the x terms as -10x instead of -14x).
Factor the expression: $ 4x^2 - 12xy + 9y^2 $
Why A is correct:
This is a perfect square trinomial. When you expand (2x - 3y)², you get (2x)² - 2(2x)(3y) + (3y)² = 4x² - 12xy + 9y². The middle term matches because -2(2x)(3y) = -12xy.
Why the others are wrong:
- B: (4x - 3y)² = 16x² - 24xy + 9y²—the first term is wrong (16x² ≠ 4x²)
- C: (x - 3y)² = x² - 6xy + 9y²—both the first term and middle term don't match
- D: (2x + 3y)² = 4x² + 12xy + 9y²—the middle term has the wrong sign (+ instead of -)
Simplify: $ \frac{2x^4y^3 - 4x^2y^2}{2x^2y^2} $
Why A is correct:
Divide each term in the numerator by the denominator separately:
- (2x^4y^3)/(2x^2y^2) = x^2y (subtract exponents: x^(4-2) = x^2, y^(3-2) = y)
- (-4x^2y^2)/(2x^2y^2) = -2 (the x^2y^2 cancels completely)
- Result: x^2y - 2 ✓
Why the others are wrong:
- B: Incorrectly uses addition (+2) instead of subtraction, and doubles the first term incorrectly
- C: Doesn't actually divide the terms; appears to subtract variables instead of simplifying
- D: Wrong coefficients and variables; doesn't follow exponent rules for division
Simplify: $ 2x[3x - 4(x - 2)] + 5 $
Why A is correct:
Work inside out: 4(x - 2) = 4x - 8, so 3x - 4(x - 2) = 3x - 4x + 8 = -x + 8. Then 2x(-x + 8) = -2x² + 16x. Finally add 5 to get -2x² + 16x + 5.
Why the others are wrong:
- B & D: These have positive x² terms, but 2x times (-x) gives negative x², not positive.
- C: The x coefficient is wrong; 2x(8) = +16x, not -16x.
Expand and simplify: $ (2x + 1)^2 - (x - 3)^2 $
Why A is correct:
- (2x + 1)² = 4x² + 4x + 1
- (x - 3)² = x² - 6x + 9
- Subtracting: (4x² + 4x + 1) - (x² - 6x + 9) = 3x² + 10x - 8 ✓
Why others are wrong:
- B: Sign error when subtracting (x - 3)²; incorrectly got -14x instead of +10x
- C: Only subtracted the x² terms but didn't fully expand or combine the linear terms correctly
- D: Made an error in expanding or combining terms; the constant should be -8, not +8
Factor completely: $ x^3 - 3x^2 - 4x + 12 $
Why A is correct:
Factor by grouping: x³ - 3x² - 4x + 12 = x²(x - 3) - 4(x - 3) = (x - 3)(x² - 4) = (x - 3)(x + 2)(x - 2). This is fully factored into three linear factors.
Why the others are wrong:
- B: Has (x + 3) instead of (x - 3). If you expand it, you get a different polynomial.
- C: This would give (x - 3)(x - 2)(x - 2), which expands to x³ - 7x² + 16x - 12—not our polynomial.
- D: Not fully factored; x² - x - 6 can still be factored into (x - 3)(x + 2), so this only completes part of the job.
Simplify: $ \frac{4x^3y - 2x^2y^2 + 6xy}{2xy} $
Why A is correct:
Divide each term in the numerator by 2xy:
- (4x³y) ÷ (2xy) = 2x²
- (−2x²y²) ÷ (2xy) = −xy
- (6xy) ÷ (2xy) = 3
Result: 2x² − xy + 3 ✓
Why the others are wrong:
- B: Has +xy instead of −xy (sign error on the second term)
- C: Has x² instead of 2x² (forgot to divide the coefficient 4 by 2)
- D: Didn't fully simplify—still has variables y in the first two terms when they should cancel out
What is the simplified form of $ (x - 1)^3 - (x + 1)^3 $?
Correct Answer: -6x^2 - 2
Expand both cubes using the formula (a ± b)³ = a³ ± 3a²b + 3ab² ± b³:
- (x - 1)³ = x³ - 3x² + 3x - 1
- (x + 1)³ = x³ + 3x² + 3x + 1
Subtract: (x³ - 3x² + 3x - 1) - (x³ + 3x² + 3x + 1) = -6x² - 2 ✓
Why others are wrong:
- B & C: These are linear expressions, but the x³ terms cancel while the x² terms don't—the result must have an x² term.
- D: Wrong sign and missing the x² term entirely; doesn't account for the expansion correctly.
Simplify: $ 2(x - 3)^2 - (x + 3)^2 $
Correct Answer: A. x^2 - 18x + 9
Expand each squared term:
- 2(x - 3)^2 = 2(x^2 - 6x + 9) = 2x^2 - 12x + 18
- (x + 3)^2 = x^2 + 6x + 9
Subtract: (2x^2 - 12x + 18) - (x^2 + 6x + 9) = x^2 - 18x + 9 ✓
Why others are wrong:
- B & C: Sign error—these have -6x or +6x instead of -18x (forgot to distribute the 2 properly or made sign mistakes)
- D: Wrong constant term—should be +9, not -9 (sign error when subtracting)
Factor: $ x^4 - 16 $
Why A is correct:
This is a difference of squares: x^4 - 16 = (x^2)^2 - 4^2, which factors as (x^2 - 4)(x^2 + 4). Then x^2 - 4 factors again as (x - 2)(x + 2), giving the complete factorization.
Why the others are wrong:
- B: (x^2 - 4)^2 expands to x^4 - 8x^2 + 16, not x^4 - 16.
- C: (x - 4)(x + 4) = x^2 - 16, not x^4 - 16 (wrong power).
- D: (x - 2)^2(x + 2)^2 expands to (x^2 - 4)^2, which is the same error as B.
Simplify: $ \frac{3x^5y^3 - 6x^3y^2 + 9x^2y}{3x^2y} $
Correct Answer (A): x^3y^2 - 2xy + 3
Divide each term in the numerator by 3x^2y:
- (3x^5y^3) ÷ (3x^2y) = x^3y^2
- (-6x^3y^2) ÷ (3x^2y) = -2xy
- (9x^2y) ÷ (3x^2y) = 3
Why others are wrong:
- B: Incorrectly simplified the middle term to -2x^2 instead of -2xy
- C: Lost an exponent on y in the first term (should be y^2, not y)
- D: Didn't divide the first term's coefficient by 3, and made errors in other terms
What is the simplified form of $ (x + y)^3 - (x - y)^3 $?
Why A is correct:
Expand both cubes using the binomial formula: (x + y)³ = x³ + 3x²y + 3xy² + y³ and (x - y)³ = x³ - 3x²y + 3xy² - y³. When you subtract them, the x³ and 3xy² terms cancel, leaving 6x²y + 2y³. Factor out 2y to get 2y(3x² + y²).
Why the others are wrong:
- B & C: These have xy as a factor, but the original expression doesn't contain an xy term after expanding and simplifying—only x² and y² terms remain.
- D: This is just a binomial with no factoring; it doesn't match the result from expanding and simplifying the cubes.
Factor completely: $ x^3 + 3x^2 - 4x - 12 $
Why A is correct:
Factor by grouping: (x³ + 3x²) + (-4x - 12) = x²(x + 3) - 4(x + 3) = (x + 3)(x² - 4). Then factor the difference of squares: x² - 4 = (x + 2)(x - 2). Final answer: (x + 3)(x + 2)(x - 2) ✓
Why the others are wrong:
- B: (x + 3)²(x - 2) expands to x³ + 4x² - 3x - 18, which doesn't match the original polynomial.
- C: (x - 3)(x + 2)² expands to x³ + x² - 8x - 12, which is incorrect.
- D: (x + 3)(x² - x - 4) expands to x³ + 2x² - 7x - 12; the x² coefficient is wrong (should be 3, not 2).
Simplify the expression: $ 2x[-3(x - 4y) + 5y] $
Correct answer: A
Work inside the brackets first: -3(x - 4y) + 5y = -3x + 12y + 5y = -3x + 17y
Then distribute the 2x: 2x(-3x + 17y) = -6x² + 34xy ✓
Why others are wrong:
- B & C: These have -22xy, which comes from errors in combining like terms inside the brackets (like forgetting to add the 5y to 12y)
- D: This has +8xy, suggesting someone only counted part of the y terms instead of combining 12y + 5y = 17y
Expand and simplify: $ (x + 2)^2 - (x - 2)^2 $
• Expand (x + 2)² = x² + 4x + 4
• Expand (x - 2)² = x² - 4x + 4
• Subtract: (x² + 4x + 4) - (x² - 4x + 4) = x² + 4x + 4 - x² + 4x - 4 = 8x ✓
Why others are wrong:
- B (4x): You'd get this if you only added one of the middle terms instead of both
- C (0): This assumes the expressions cancel completely—they don't
- D (4): This ignores the x terms entirely
Factor completely: $ 6x^2 - 15x $
Why A is correct:
Find the GCF of both terms: 6x² and 15x share 3x as their greatest common factor. Factor it out: 3x(2x − 5). Check: 3x · 2x = 6x² ✓ and 3x · (−5) = −15x ✓
Why the others are wrong:
- B: Missing the x factor—you'd get 3(2x − 5) = 6x − 15, which loses the x² term entirely.
- C: The GCF is 3x, not 3x². Also, 3x²(2 − 5x) = 6x² − 15x³, which is wrong.
- D: While technically correct, it's not *completely* factored because 6x can be factored further into 3x · 2.
Divide and simplify: $ \frac{10x^4y^3}{2x^2y} $
A is correct. Divide the numbers: 10 ÷ 2 = 5. Divide the x terms using exponent rules: x^4 ÷ x^2 = x^(4-2) = x^2. Divide the y terms: y^3 ÷ y = y^(3-1) = y^2. This gives 5x^2y^2.
B is wrong: This has x^3 instead of x^2—you'd get this if you subtracted exponents incorrectly (4 - 2 ≠ 3).
C is wrong: Multiplying instead of dividing the coefficients (10 × 2 = 20) is a common mistake.
D is wrong: This adds the y exponents (3 + 1 = 4) instead of subtracting them.
Simplify: $ (2x - 3)^2 $
Why A is correct:
Use the formula (a - b)² = a² - 2ab + b². Here: (2x)² = 4x², then -2(2x)(3) = -12x, then (-3)² = 9. Combined: 4x² - 12x + 9.
Why the others are wrong:
- B: Ignores the middle term (-12x) entirely; just squares each piece separately.
- C: Incorrectly squares the coefficient (2² = 4, not 2) and doesn't account for the middle term.
- D: Makes the middle term too small (-6x instead of -12x); this misses the factor of 2 in the middle term formula.
Simplify: $ -4[2x - 3(x + y)] $
Why A is correct:
Work inside out: 3(x + y) = 3x + 3y, so 2x - 3(x + y) = 2x - 3x - 3y = -x - 3y. Then -4(-x - 3y) = 4x + 12y ✓
Why the others are wrong:
- B: Forgot to distribute the negative sign properly through both terms
- C: Made an error in the inner simplification or didn't fully distribute the -4
- D: This would be correct if the outer coefficient were +2, not -4
Expand: $ (x + 1)(x^2 - x + 3) $
Why A is correct:
Distribute each term in (x + 1) across (x² - x + 3):
- x · x² = x³
- x · (-x) = -x²
- x · 3 = 3x
- 1 · x² = x²
- 1 · (-x) = -x
- 1 · 3 = 3
Combine: x³ + (-x² + x²) + (3x - x) + 3 = x³ + 0x² + 2x + 3 ✓
Why others are wrong:
- B: Incorrectly combined like terms (got -x² instead of 0x²)
- C: Sign errors in distribution and combining terms
- D: Missing the -x² and -x terms entirely; incomplete expansion
Simplify: $ \frac{4x^3y^2}{2xy} $
A is correct: $2x^2y$
Divide the coefficients: 4 ÷ 2 = 2. Then use exponent rules to subtract powers: $x^3 ÷ x = x^{3-1} = x^2$ and $y^2 ÷ y = y^{2-1} = y^1 = y$. Result: $2x^2y$.
Why the others are wrong:
- B ($2x^2y^2$): Incorrectly kept $y^2$ instead of subtracting the exponents.
- C ($4x^2y$): Forgot to divide the coefficients—used 4 instead of 4 ÷ 2 = 2.
- D ($2xy^2$): Made two mistakes: didn't fully simplify the $x$ terms and kept $y^2$ instead of $y$.
Factor: $ x^2 - 2x - 15 $
Why A is correct:
You need two numbers that multiply to -15 and add to -2. Those numbers are -5 and +3: (-5)(+3) = -15 and -5 + 3 = -2. So the factors are (x - 5)(x + 3).
Why the others are wrong:
- B: (x - 3)(x + 5) gives -3 and +5, which add to +2, not -2.
- C: (x + 5)(x + 3) gives +5 and +3, which multiply to +15, not -15.
- D: (x - 1)(x - 15) gives -1 and -15, which add to -16, not -2.
Expand: $ (2x - 1)^2 $
Why A is correct:
Use the formula (a - b)² = a² - 2ab + b². Here: (2x)² = 4x², then -2(2x)(1) = -4x, then (-1)² = 1. Result: 4x² - 4x + 1.
Why others are wrong:
- B: Incorrectly calculates the middle term as -x instead of -4x (forgot to multiply by 2).
- C: Same error—middle term should be -4x, not -2x.
- D: Wrong first term; (2x)² = 4x², not 2x².
Simplify: $ 3x^2 - 2x + 5 + x^2 + 4x - 3 $
Correct answer: A ($4x^2 + 2x + 2$)
Combine like terms:
- x² terms: 3x² + x² = 4x²
- x terms: -2x + 4x = 2x
- Constants: 5 + (-3) = 2
Result: 4x² + 2x + 2
Why the others are wrong:
- B: Only adds 1x² instead of 2, giving 3x² (arithmetic error)
- C: Combines the x terms wrong (adds them as -2x + 4x + something extra = 6x instead of 2x)
- D: Adds the constants incorrectly as 5 + 3 = 8 instead of 5 - 3 = 2
Factor: $ 9x^2 - 25 $
A is correct because this is a difference of squares: 9x² = (3x)² and 25 = 5², so 9x² - 25 = (3x)² - 5² = (3x - 5)(3x + 5).
B is wrong because (3x - 5)² expands to 9x² - 30x + 25, which has a middle term and doesn't match our original expression.
C is wrong because (x - 5)(x + 5) = x² - 25, not 9x² - 25.
D is wrong because we just factored it—it absolutely can be factored as a difference of squares.
Expand: $ (x + 3)(x - 3) $
Why A is correct:
This is the difference of squares pattern: (a + b)(a - b) = a² - b². Here, a = x and b = 3, so you get x² - 3² = x² - 9.
Why the others are wrong:
- B ($x^2 + 6x + 9$): This is (x + 3)² — you'd only get this if you multiplied (x + 3) by itself, not by (x - 3).
- C ($x^2 - 6x + 9$): This is (x - 3)² — same mistake as B, but with the subtraction version.
- D ($x^2 + 9$): This forgets the middle term entirely; the difference of squares always produces a subtraction in the result.
Divide: $ \frac{-12a^3b^2}{4ab} $
Correct answer: A. $-3a^2b$
Divide each part separately:
- Coefficients: $-12 ÷ 4 = -3$ ✓
- Variable $a$: $a^3 ÷ a = a^2$ (subtract exponents: $3-1=2$) ✓
- Variable $b$: $b^2 ÷ b = b$ (subtract exponents: $2-1=1$) ✓
Why others are wrong:
- B ($-3ab^2$): Forgot to subtract the exponent of $a$ (should be $a^2$, not $a$)
- C ($-4a^2b^2$): Wrong coefficient ($-4$ instead of $-3$) AND didn't subtract the $b$ exponent
- D ($-2a^2b$): Wrong coefficient ($-2$ instead of $-3$)
Factor: $ x^2 + 6x + 9 $
Correct answer: (x + 3)²
This is a perfect square trinomial. When you expand (x + 3)², you get x² + 6x + 9 ✓
Why the others are wrong:
- (x + 9)(x + 1): Expands to x² + 10x + 9 (wrong middle term)
- (x + 2)²: Expands to x² + 4x + 4 (wrong middle and last terms)
- (x + 6)²: Expands to x² + 12x + 36 (wrong middle and last terms)
Expand and simplify: $ (3x + 2)^2 $
Why A is correct:
Use the formula (a + b)² = a² + 2ab + b². Here, a = 3x and b = 2, so:
- (3x)² = 9x²
- 2(3x)(2) = 12x
- 2² = 4
- Result: 9x² + 12x + 4 ✓
Why the others are wrong:
- B (6x² + 4x + 4): Forgot to square 3x properly; (3x)² = 9x², not 6x²
- C (9x² + 6x + 4): Got the first and last terms right, but the middle term should be 12x (double the product), not 6x
- D (9x² + 12x + 2): Correct middle term, but squared the constant wrong; 2² = 4, not 2
Simplify: $ 4x^2 - [2x^2 - 3x + 5] $
Correct Answer: A
When you subtract a bracketed expression, distribute the negative sign to every term inside: 4x² - [2x² - 3x + 5] = 4x² - 2x² + 3x - 5 = 2x² + 3x - 5.
Why others are wrong:
- B: Adds the x² terms instead of subtracting (4x² + 2x² = 6x²)
- C: Forgets to distribute the negative sign to -3x and +5 (keeps them unchanged)
- D: Distributes the negative correctly but then gets the constant wrong (should be -5, not +5)
Factor: $ x^2 - 10x + 25 $
A. $(x - 5)^2$ is correct.
This is a perfect square trinomial. When you expand $(x - 5)^2 = (x - 5)(x - 5) = x^2 - 10x + 25$ ✓. The pattern is: first term squared, minus twice the product of both terms, plus last term squared.
Why the others are wrong:
- B. $(x - 5)(x + 5) = x^2 - 25$ (difference of squares, not this expression)
- C. $(x - 10)(x + 2.5) = x^2 - 7.5x - 25$ (doesn't match)
- D. $(x - 2)^2 = x^2 - 4x + 4$ (wrong middle term and constant)
Expand: $ (x - 1)(x^2 + x + 1) $
Why A is correct:
This is the difference of cubes formula: (a - b)(a² + ab + b²) = a³ - b³. Here a = x and b = 1, so (x - 1)(x² + x + 1) = x³ - 1.
Why the others are wrong:
- B & D: Missing or misplace terms. If you multiply out fully, you'd see x³ terms cancel with negative x³ terms, leaving only x³ - 1.
- C: This results from incorrectly distributing; it has too many middle terms that should cancel out during multiplication.
Divide and simplify: $ \frac{15x^4y^3}{5x^2y} $
A is correct: $3x^2y^2$
• Divide the coefficients: 15 ÷ 5 = 3
• Divide the x terms: $x^4 ÷ x^2 = x^2$ (subtract exponents: 4 − 2 = 2)
• Divide the y terms: $y^3 ÷ y = y^2$ (subtract exponents: 3 − 1 = 2)
Why others are wrong:
- B ($3x^3y^2$): Wrong exponent on x—didn't subtract correctly (should be 4 − 2 = 2, not 3)
- C ($10x^2y^2$): Wrong coefficient—subtracted instead of dividing (15 − 5 ≠ 15 ÷ 5)
- D ($3x^2y^4$): Wrong exponent on y—added instead of subtracting (3 + 1 ≠ 3 − 1)
Expand: $ (x + 5)(x - 2) $
A is correct.
Use FOIL (First, Outer, Inner, Last):
- First: x · x = x²
- Outer: x · (–2) = –2x
- Inner: 5 · x = 5x
- Last: 5 · (–2) = –10
Combine like terms: x² + (–2x + 5x) – 10 = x² + 3x – 10
Why others are wrong:
- B: Wrong middle term sign (–3x instead of +3x)
- C: Wrong middle term coefficient (10x instead of 3x)
- D: Wrong last term sign (+10 instead of –10)
Simplify: $ 2x(3x - 4y + 5) + 6y $
Correct Answer: A
Distribute 2x to each term inside the parentheses: 2x(3x) = 6x², 2x(−4y) = −8xy, and 2x(5) = 10x. Then add the 6y outside: 6x² − 8xy + 10x + 6y.
Why others are wrong:
- B: Sign error on the final term—it's +6y, not −6y.
- C: Missing the −8xy term from distributing 2x(−4y).
- D: Missing the 10x term from distributing 2x(5).
Factor: $ x^2 - 6x + 9 $
A. $(x - 3)^2$ ✓
This is a perfect square trinomial: $x^2 - 6x + 9 = (x-3)(x-3) = (x-3)^2$. Check: the first term is $x^2$, the last term is $3^2 = 9$, and the middle term is $-2(x)(3) = -6x$. ✓
B. $(x - 3)(x + 3)$ ✗
This is the difference of squares pattern, which gives $x^2 - 9$, not $x^2 - 6x + 9$.
C. $(x - 6)(x + 1.5)$ ✗
Expanding this gives $x^2 - 4.5x - 9$, which doesn't match our original expression.
D. $(x - 2)^2$ ✗
This expands to $x^2 - 4x + 4$, which has the wrong middle and last terms.
Expand and simplify: $ (x - 4)(x + 2) $
A is correct.
Use FOIL to expand: First (x · x = x²), Outer (x · 2 = 2x), Inner (-4 · x = -4x), Last (-4 · 2 = -8).
Combine like terms: x² + 2x - 4x - 8 = x² - 2x - 8 ✓
Why the others are wrong:
- B: Signs error—added 2x instead of combining 2x and -4x correctly
- C: Wrong middle term; this would come from (x + 4)(x + 2)
- D: Wrong signs; the middle term should be negative -2x, not -6x
Divide and simplify: $ \frac{18x^5y^2}{3x^2y} $
• Divide the coefficients: 18 ÷ 3 = 6 ✓
• Divide the x terms: x^5 ÷ x^2 = x^(5-2) = x^3 ✓
• Divide the y terms: y^2 ÷ y^1 = y^(2-1) = y^1 = y ✓
• Result: 6x³y
Why others are wrong:
- B (6x²y²): Used wrong exponents—subtracted incorrectly for x and didn't simplify y properly
- C (6x³y²): Got x right but forgot to subtract the y exponents (should be y¹, not y²)
- D (15x³y): Added coefficients instead of dividing (18 + 3 ≠ correct operation)
Simplify: $ 2(3x - 4) - 5(x + 1) $
Correct answer: A ($x - 13$)
Distribute each term: 2(3x - 4) = 6x - 8 and -5(x + 1) = -5x - 5. Combine them: 6x - 8 - 5x - 5 = x - 13. ✓
Why the others are wrong:
- B ($x + 13$): Sign error—you'd need to add the constant terms instead of subtract them.
- C ($x - 9$): Arithmetic mistake combining constants: -8 - 5 = -13, not -9.
- D ($-x - 13$): Sign error combining x-terms: 6x - 5x = +x, not -x.
Factor: $ 4x^2 - 12x + 9 $
A. $(2x - 3)^2$ ✓ CORRECT
This is a perfect square trinomial. When you expand $(2x - 3)^2 = (2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9$. Perfect match!
B. $(2x + 3)^2$
Expanding gives $4x^2 + 12x + 9$—the middle term is positive, but we need negative 12x.
C. $(4x - 3)(x - 3)$
This expands to $4x^2 - 15x + 9$—the middle term is wrong (−15x instead of −12x).
D. $(x - 3)^2$
This gives $x^2 - 6x + 9$—completely different coefficients; doesn't match our expression.
Expand: $ (2x - 1)(x + 4) $
Why A is correct:
Use FOIL: (2x)(x) = 2x², (2x)(4) = 8x, (−1)(x) = −x, (−1)(4) = −4. Combine: 2x² + 8x − x − 4 = 2x² + 7x − 4. ✓
Why others are wrong:
- B (2x² + 3x − 4): Middle term is wrong; combining 8x and −x gives 7x, not 3x.
- C (2x² + 5x − 4): Same issue—forgot to properly combine the middle terms.
- D (2x² − 7x − 4): Sign error; the middle term should be positive +7x, not negative.
Simplify: $ \frac{-10x^3y^2}{5xy} $
Correct answer (A): $-2x^2y$
Divide the coefficients: $-10 ÷ 5 = -2$. Then use the exponent rule (subtract powers when dividing): $x^3 ÷ x^1 = x^2$ and $y^2 ÷ y^1 = y^1$. This gives $-2x^2y$.
Why the others are wrong:
- B ($-2xy^2$): The $y$ exponents weren't simplified correctly—you subtract 1 from 2, not keep it as 2.
- C ($-5x^2y^2$): The coefficient should be $-2$, not $-5$; you divide by 5, not multiply.
- D ($-2x^3y^2$): The $x$ exponent wasn't simplified—it should be $x^2$, not $x^3$.
Factor: $ x^2 - 4x - 21 $
Why A is correct:
You need two numbers that multiply to -21 and add to -4. Those numbers are -7 and +3: (-7)(+3) = -21 and -7 + 3 = -4. So the factors are (x - 7)(x + 3).
Why the others are wrong:
- B: (7)(-3) = -21 ✓, but 7 + (-3) = 4, not -4 ✗
- C: (7)(3) = 21, not -21 ✗
- D: (-3)(-7) = 21, not -21 ✗
Simplify: $ 3x(2x - 5y + 4) + 2y(3x - y) $
# Explanation
Why A is correct:
Distribute 3x into the first group: 3x(2x) = 6x², 3x(-5y) = -15xy, 3x(4) = 12x.
Distribute 2y into the second group: 2y(3x) = 6xy, 2y(-y) = -2y².
Combine like terms: 6x² + (-15xy + 6xy) + 12x - 2y² = 6x² - 9xy + 12x - 2y² ✓
Why the others are wrong:
- B: Sign error—incorrectly uses +15xy instead of combining -15xy + 6xy, and +2y² instead of -2y².
- C: Missing the 12x term (forgot to distribute 3x · 4) and incorrectly combined the xy terms.
- D: Made an arithmetic error combining xy terms; should get -9xy, not -15xy (forgot to add the +6xy from the second group).
Factor completely: $ x^2 - 2x - 35 $
Why A is correct:
You need two numbers that multiply to -35 and add to -2. Those numbers are -7 and +5: (-7)(+5) = -35 and -7 + 5 = -2. So the factors are (x - 7)(x + 5).
Why the others are wrong:
- B: (-5)(+7) = -35 ✓, but -5 + 7 = +2 ✗ (wrong middle term)
- C: (+7)(+5) = +35 ✗ (wrong sign) and +7 + 5 = +12 ✗
- D: (-7)(-5) = +35 ✗ (wrong sign) and -7 - 5 = -12 ✗
Expand: $ (x + 3)^2 - (x - 2)^2 $
• Correct answer (A): Expand each square separately: (x + 3)² = x² + 6x + 9 and (x - 2)² = x² - 4x + 4. Subtract them: (x² + 6x + 9) - (x² - 4x + 4) = x² + 6x + 9 - x² + 4x - 4 = 10x + 5.
• Why B, C, D are wrong: They all have incorrect coefficients or constants. B reverses the coefficient and constant (5x + 10). C and D have much smaller coefficients, suggesting incomplete expansion or arithmetic errors when combining like terms.
Divide: $ \frac{12a^3b^2}{4ab} $
Why A is correct:
Divide the coefficients: 12 ÷ 4 = 3. For variables, subtract exponents: a³ ÷ a = a² and b² ÷ b = b. Result: 3a²b
Why others are wrong:
- B (3ab²): You divided the b's incorrectly—b² ÷ b = b, not b²
- C (4a²b²): Wrong coefficient (should be 3, not 4) and didn't divide the b's
- D (3a³b): Forgot to subtract the exponent on a—should be a², not a³
Simplify: $ (x + 2)(x - 2) $
A is correct: This is a difference of squares pattern: (a + b)(a - b) = a² - b². Here, a = x and b = 2, so you get x² - 2² = x² - 4. You can verify by FOILing: x² - 2x + 2x - 4 = x² - 4.
B is wrong: You can't add the squares; the middle terms (-2x and +2x) cancel out, leaving subtraction.
C and D are wrong: These come from incorrectly expanding or mixing up signs. They have extra terms that don't appear when you properly multiply or use the difference of squares rule.
Factor the expression: $ 8x^2 - 2x $
Why A is correct:
Factor out the greatest common factor (GCF) of both terms. Both 8x² and 2x share 2x, so: 8x² - 2x = 2x(4x - 1). Check by expanding: 2x · 4x - 2x · 1 = 8x² - 2x ✓
Why the others are wrong:
- B: Only factors out 2, not the full GCF. You'd still have x² and x inside the parentheses, so it's not completely factored.
- C: Has the wrong sign—should be minus, not plus. Expanding gives 2x(4x + 1) = 8x² + 2x, which doesn't match.
- D: Factors out 4x (too much). Expanding gives 4x(2x - 1) = 8x² - 4x, which is incorrect.
Simplify: $ 5x^2 + 3x - 2x^2 - 4x $
Correct answer: A. $3x^2 - x$
- Combine like terms: the $x^2$ terms are $5x^2 - 2x^2 = 3x^2$, and the $x$ terms are $3x - 4x = -x$
- Result: $3x^2 - x$ ✓
Why others are wrong:
- B ($3x^2 + x$): Wrong sign on the $x$ term; you'd get this if you added instead of subtracted the $x$ coefficients
- C ($7x^2 - x$): Adds the $x^2$ terms instead of subtracting them
- D ($3x^2 - 7x$): Correctly finds $3x^2$, but adds all the $x$ coefficients instead of combining them properly
Factor: $ x^2 + 6x + 8 $
Why A is correct:
When you expand (x + 4)(x + 2), you get x² + 2x + 4x + 8 = x² + 6x + 8. Also, 4 and 2 are two numbers that multiply to 8 and add to 6, which is exactly what you need.
Why the others are wrong:
- B: (x + 3)(x + 3) = x² + 6x + 9, not x² + 6x + 8 (the constant is wrong)
- C: (x + 4)(x - 2) = x² + 2x - 8, not x² + 6x + 8 (wrong middle and constant terms)
- D: (x + 2)² = x² + 4x + 4, not x² + 6x + 8 (completely different)
Simplify: $ 4a^2b - 2ab^2 + 3a^2b - ab^2 $
Correct answer: A
Combine like terms by grouping terms with the same variables and powers:
- $a^2b$ terms: $4a^2b + 3a^2b = 7a^2b$
- $ab^2$ terms: $-2ab^2 - ab^2 = -3ab^2$
- Result: $7a^2b - 3ab^2$
Why others are wrong:
- B: Only combined one $ab^2$ term instead of both ($-2ab^2 - ab^2 = -3ab^2$, not $-ab^2$)
- C: Added the $ab^2$ terms instead of subtracting them
- D: Flipped which variable has the larger exponent; the $a^2b$ terms are larger
Simplify: $ \frac{12x^3y^2}{4xy} $
• Divide the coefficients: 12 ÷ 4 = 3 ✓
• Divide the variables using exponent rules: x³ ÷ x = x^(3-1) = x² and y² ÷ y = y^(2-1) = y ✓
• Result: 3x²y
Why others are wrong:
- B (3x³y²): Forgot to subtract exponents; just kept the original powers
- C (3x²y²): Correctly simplified x, but forgot to reduce y² ÷ y to y (not y²)
- D (3x²): Got the x terms right but completely dropped the y term
Expand: $ (x + 2)(x - 5) $
A is correct. Use FOIL: First (x·x = x²), Outer (x·−5 = −5x), Inner (2·x = 2x), Last (2·−5 = −10). Combine like terms: x² − 5x + 2x − 10 = x² − 3x − 10.
B is wrong: This has +3x instead of −3x—you'd get this if you accidentally added 5x + 2x instead of −5x + 2x.
C is wrong: This has −7x, which would only happen if you incorrectly combined the outer and inner terms as −5x − 2x (forgetting the sign on the 2).
D is wrong: The constant term should be negative (2 times −5 = −10, not +10).
Simplify: $ (3x - 4)^2 $
Why A is correct:
Use the formula (a - b)² = a² - 2ab + b². Here: (3x)² = 9x², then -2(3x)(4) = -24x, then 4² = 16. So you get 9x² - 24x + 16.
Why others are wrong:
- B: Uses -12x instead of -24x (forgot to multiply by 2 in the middle term)
- C: Has +24x instead of -24x (wrong sign on the middle term)
- D: Has 3x² instead of 9x² (forgot to square the coefficient 3)
Factor: $ x^2 - 9 $
Why A is correct:
This is the difference of squares pattern: x² - 9 = x² - 3². The formula is a² - b² = (a - b)(a + b), so x² - 9 = (x - 3)(x + 3). Check: (x - 3)(x + 3) = x² - 9 ✓
Why the others are wrong:
- B: (x - 9)(x + 1) = x² - 8x - 9, not x² - 9
- C: (x - 3)² = x² - 6x + 9, not x² - 9 (this is a perfect square, not a difference of squares)
- D: (x + 9)(x - 1) = x² + 8x - 9, not x² - 9
Factor completely: $ x^2 + 2x - 15 $
Why A is correct:
You need two numbers that multiply to –15 and add to +2. That's +5 and –3: (5)(–3) = –15 and 5 + (–3) = 2. So the factors are (x + 5)(x – 3).
Why the others are wrong:
- B: (–5)(+3) = –15 ✓, but –5 + 3 = –2, not +2 ✗
- C: (x – 5)² expands to x² – 10x + 25, not x² + 2x – 15 ✗
- D: (+3)(+5) = +15, not –15 ✗
Simplify: $ 4x^2 - [3x^2 - 2x + 7] $
Correct Answer: A ($x^2 + 2x - 7$)
When you subtract a bracketed expression, distribute the negative sign to every term inside: 4x² - 3x² + 2x - 7 = x² + 2x - 7.
Why others are wrong:
- B has -2x instead of +2x (forgot to distribute the negative to -2x)
- C has +7 instead of -7 (didn't distribute the negative to the 7)
- D has both errors combined (wrong signs on both the x and constant terms)
Expand: $ (x + 1)(x^2 - x + 1) $
A. $x^3 + 1$ is correct.
Multiply each term in (x + 1) by each term in (x² - x + 1):
- x · (x² - x + 1) = x³ - x² + x
- 1 · (x² - x + 1) = x² - x + 1
- Add them: x³ - x² + x + x² - x + 1 = x³ + 1 (the middle terms cancel)
This is actually the sum of cubes formula: (a + b)(a² - ab + b²) = a³ + b³.
Why the others are wrong:
- B, C, D all have extra terms that shouldn't survive the cancellation. When you combine like terms properly, all the x² and x terms eliminate, leaving only x³ + 1.
Simplify: $ \frac{6a^4b^2}{3ab} $
Correct Answer: A ($2a^3b$)
Divide the coefficients: 6 ÷ 3 = 2. For the variables, subtract exponents when dividing: a^4 ÷ a^1 = a^3, and b^2 ÷ b^1 = b^1. This gives you 2a^3b.
Why the others are wrong:
- B ($2a^3b^2$): You correctly divided the a's but forgot to subtract the b exponents—b^2 ÷ b should give b^1, not b^2.
- C ($3a^3b$): You correctly simplified the variables but made an error dividing coefficients (6 ÷ 3 = 2, not 3).
- D ($2a^2b$): You got the coefficient right but subtracted the exponents incorrectly—a^4 ÷ a gives a^3, not a^2.
Factor: $ x^2 + 10x + 21 $
Correct answer: (x + 3)(x + 7)
When you multiply this out: (x + 3)(x + 7) = x² + 7x + 3x + 21 = x² + 10x + 21 ✓
You need two numbers that multiply to 21 AND add to 10. That's 3 and 7.
Why the others are wrong:
- B. (x + 4)(x + 6): These multiply to 24, not 21, and add to 10. Close but wrong product.
- C. (x + 2)(x + 10): These multiply to 20, not 21. Also adds to 12, not 10.
- D. (x - 3)(x - 7): This gives x² - 10x + 21 (negative middle term). Wrong sign.
Expand: $ (2x - 3)(x^2 + x - 4) $
Why A is correct:
Distribute each term in (2x - 3) across (x² + x - 4):
- 2x · x² = 2x³
- 2x · x = 2x²
- 2x · (-4) = -8x
- (-3) · x² = -3x²
- (-3) · x = -3x
- (-3) · (-4) = +12
Combine like terms: 2x³ + (2x² - 3x²) + (-8x - 3x) + 12 = 2x³ - x² - 11x + 12 ✓
Why others are wrong:
- B: Wrong x² term (should be negative) and wrong x coefficient
- C: Missing the x² term entirely and has wrong x coefficient
- D: Wrong x² term and wrong x coefficient (positive instead of negative)
In the term $-7ab$, which of the following lists all its factors?
Why A is correct:
The factors of $-7ab$ are all the pieces that multiply together to make this term. Since $-7ab = (-7) \times a \times b$, the complete list of factors is $-7, a, b$.
Why the others are wrong:
- B: Missing the negative sign. The coefficient is $-7$, not $7$.
- C: Missing $b$. We need all three factors to fully break down the term.
- D: Missing $a$. We need all three factors to fully break down the term.
Simplify: $-2[-3(x - 2y) + 4y]$.
Why A is correct:
Work from inside the brackets outward: $-3(x - 2y) + 4y = -3x + 6y + 4y = -3x + 10y$. Then multiply by $-2$: $-2(-3x + 10y) = 6x - 20y$.
Why the others are wrong:
- B ($-6x + 20y$): Sign error—you'd get this if you forgot to distribute the negative sign from $-2$.
- C ($6x + 20y$): Wrong sign on the $y$ term; happens if you make an error combining like terms inside the brackets.
- D ($-6x - 20y$): Both signs are wrong; this comes from not properly multiplying by $-2$ at the end.
Simplify: $5x[-4 + 10(x - y)] + 7x$.
Why A is correct:
Work inside the brackets first: $10(x-y) = 10x - 10y$, so $[-4 + 10x - 10y]$.
Then distribute the $5x$: $5x(-4) + 5x(10x) + 5x(-10y) = -20x + 50x^2 - 50xy$.
Finally add $7x$: $-20x + 7x + 50x^2 - 50xy = 50x^2 - 13x - 50xy$.
Why the others are wrong:
- B: Forgot to combine $-20x + 7x$; kept them separate instead of simplifying to $-13x$.
- C: Made a sign error, getting $-7x$ instead of $-13x$ when combining like terms.
- D: Didn't distribute and combine properly; left the expression incomplete.
Multiply: $(a + b)(x + y)$.
Why A is correct:
Use the distributive property (FOIL): multiply each term in the first parentheses by each term in the second. You get $a·x + a·y + b·x + b·y = ax + ay + bx + by$.
Why the others are wrong:
- B ($ax + by$): Missing two terms; you forgot to multiply $a·y$ and $b·x$.
- C ($ax + by$): Same problem as B—incomplete distribution.
- D ($ab + xy$): This multiplies within each parenthesis instead of across them; the distributive property requires multiplying *every* term in one set by *every* term in the other.
Expand and simplify: $(2x + 3)(x^2 - x - 5)$.
# Explanation
Why A is correct:
Use the distributive property (FOIL for binomials × trinomials): multiply each term in $(2x + 3)$ by each term in $(x^2 - x - 5)$.
- $2x \cdot x^2 = 2x^3$
- $2x \cdot (-x) = -2x^2$
- $2x \cdot (-5) = -10x$
- $3 \cdot x^2 = 3x^2$
- $3 \cdot (-x) = -3x$
- $3 \cdot (-5) = -15$
Combine: $2x^3 + (-2x^2 + 3x^2) + (-10x - 3x) - 15 = 2x^3 + x^2 - 13x - 15$ ✓
Why others are wrong:
- B: Wrong $x^2$ coefficient (should be $+1$, not $-1$) and wrong $x$ coefficient.
- C: Wrong $x^2$ coefficient ($+3$ instead of $+1$) and wrong $x$ coefficient.
- D: Wrong $x$ coefficient ($-8$ instead of $-13$); the middle terms $-2x^2 + 3x^2 = x^2$ weren't combined correctly here.
Expand and simplify: $(x - 3)^2$.
Why A is correct:
Use the formula $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = x$ and $b = 3$, so: $x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9$.
Why the others are wrong:
- B: Has $+6x$ instead of $-6x$—this would be $(x + 3)^2$.
- C: This is the difference of squares formula $x^2 - 9$, which only works for $(x-3)(x+3)$, not $(x-3)^2$.
- D: Multiplies incorrectly; the middle term should be $-2(3x) = -6x$, not $-3x$.
Expand and simplify: $(a - 2y^2)^2$.
Correct answer (A): $a^2 - 4ay^2 + 4y^4$
Use the formula $(x - y)^2 = x^2 - 2xy + y^2$ with $x = a$ and $y = 2y^2$:
- First term: $a^2$ ✓
- Middle term: $-2(a)(2y^2) = -4ay^2$ ✓
- Last term: $(2y^2)^2 = 4y^4$ ✓
Why others are wrong:
- B: The middle coefficient should be $-4$, not $-2$. (Forgot to multiply by 2 in the formula.)
- C: The middle term is positive, but $(a - 2y^2)^2$ has a negative middle term since we're subtracting.
- D: The middle term has $a^2y^2$, not $ay^2$. This comes from incorrectly squaring the $a$ in the middle term.
Combine like terms: $2x^3 + 5x^3$.
• A is correct: Both terms have the same variable and exponent ($x^3$), so they're "like terms." Add the coefficients: 2 + 5 = 7, keeping $x^3$ unchanged. Result: $7x^3$.
• B is wrong: You might get this if you mistakenly multiplied the coefficients (2 × 5 = 10) instead of adding them.
• C is wrong: This incorrectly adds the exponents (3 + 3 = 6). You never add exponents when combining like terms—only add the coefficients.
• D is wrong: This ignores the coefficients entirely. The coefficient isn't 1; it's 2 + 5 = 7.
Simplify: $\frac{3x^7}{x^2}$.
• Why A is correct: When dividing powers with the same base, subtract the exponents: $\frac{3x^7}{x^2} = 3 \cdot x^{7-2} = 3x^5$. The coefficient 3 stays in the numerator.
• Why B is wrong: This puts 3 in the denominator, but 3 is already in the numerator with no x attached—it doesn't get divided.
• Why C is wrong: This adds the exponents ($7+2$) instead of subtracting them. That's what you do when *multiplying* powers, not dividing.
• Why D is wrong: This is the original unsimplified expression rewritten as multiplication—it doesn't simplify anything.
Simplify: $\frac{-6a^2bc^2}{2abc^4}$.
Correct answer (A):
Divide the coefficients: $-6 ÷ 2 = -3$. Cancel matching variables: $a^2 ÷ a = a$, $b ÷ b = 1$, and $c^2 ÷ c^4 = 1/c^2$. This gives $(-3a)/(c^2)$.
Why the others are wrong:
- B: Keeps an extra $b$ in the numerator, but $b$ cancels completely ($b ÷ b = 1$).
- C: Doesn't fully simplify the coefficient—$-6/2 = -3$, not $-6/2$.
- D: Doesn't simplify the $b$ terms or the coefficient, leaving unnecessary terms.
Simplify: $\frac{(x+4)(x-3)}{(x-3)}$.
Correct answer: A. $x+4$
• You can cancel the $(x-3)$ factor in the numerator with the $(x-3)$ factor in the denominator, leaving just $x+4$.
• This works because any non-zero number divided by itself equals 1: $\frac{x-3}{x-3} = 1$ (as long as $x \neq 3$).
Why the others are wrong:
• B ($x-3$): This is the factor you *cancel out*, not the answer.
• C ($1$): This would only be right if the entire numerator and denominator were equal, but they're not.
• D (unchanged fraction): This is what you get if you forget to simplify—you haven't actually done anything.
Simplify: $3[2x - (x + 3)] - 4x$.
• Work inside out: First simplify $2x - (x + 3) = 2x - x - 3 = x - 3$.
• Distribute the 3: $3(x - 3) - 4x = 3x - 9 - 4x$.
• Combine like terms: $3x - 4x - 9 = -x - 9$. ✓
Why others are wrong:
- B ($x - 9$): Sign error when combining $3x - 4x$.
- C ($-x + 9$): Wrong sign on the constant (should be $-9$, not $+9$).
- D ($3x - 4x - 3$): Stopped too early without combining terms or made an error distributing the 3.
Expand and simplify: $(x + y)^2$.
Why A is correct:
$(x + y)^2 = (x + y)(x + y)$. Using FOIL: First ($x \cdot x = x^2$), Outer ($x \cdot y = xy$), Inner ($y \cdot x = xy$), Last ($y \cdot y = y^2$). Combining the middle terms: $x^2 + 2xy + y^2$.
Why the others are wrong:
- B ($x^2 + y^2$): Missing the middle term $2xy$ entirely. This is a common mistake when students forget to multiply the cross terms.
- C ($x^2 + xy + y^2$): Only counts one of the two $xy$ terms. You get $xy$ twice when you expand, not once.
- D ($x^2 - 2xy + y^2$): This is the expansion of $(x - y)^2$, not $(x + y)^2$. The sign is wrong.
Expand and simplify: $(2x - y)^2$.
A is correct. Use the formula $(a - b)^2 = a^2 - 2ab + b^2$. Here: $(2x)^2 - 2(2x)(y) + y^2 = 4x^2 - 4xy + y^2$.
B is wrong: This has a plus sign in the middle term, but $(2x - y)^2$ has a minus sign, so the middle term must be negative.
C is wrong: The first term should be $(2x)^2 = 4x^2$, not $2x^2$. The middle term is also too small.
D is wrong: The middle term is $-2xy$, but it should be $-2(2x)(y) = -4xy$. The coefficient got cut in half.
Factor completely: $9x^2 -24xy +16y^2$.
Why A is correct:
This is a perfect square trinomial. Check: $(3x-4y)^2 = 9x^2 - 2(3x)(4y) + 16y^2 = 9x^2 - 24xy + 16y^2$ ✓
Why the others are wrong:
- B: $(3x+4y)^2 = 9x^2 + 24xy + 16y^2$ — the middle term is positive, but we need negative
- C: $(9x-16y)^2 = 81x^2 - 288xy + 256y^2$ — wrong coefficients entirely
- D: $(3x-4y)(3x+4y) = 9x^2 - 16y^2$ — this is a difference of squares with no middle term
Factor completely: $4x^2 -12xy +9y^2$.
A is correct: This is a perfect square trinomial. Check: $(2x-3y)^2 = 4x^2 - 12xy + 9y^2$ ✓. The pattern is $a^2 - 2ab + b^2 = (a-b)^2$ where $a=2x$ and $b=3y$.
B is wrong: $(2x+3y)^2 = 4x^2 + 12xy + 9y^2$—this has a plus middle term, but our expression has minus 12xy.
C is wrong: $(4x-9y)^2 = 16x^2 - 72xy + 81y^2$—completely different coefficients; this doesn't match.
D is wrong: This is difference of squares pattern $(a-b)(a+b) = a^2 - b^2$, which gives $4x^2 - 9y^2$ (no middle term at all).
Simplify: $\frac{6p^3q^2}{2pq}$.
• Correct (A): Divide the coefficients: 6÷2 = 3. For the variables, subtract exponents when dividing: p³÷p = p² and q²÷q = q¹. Result: 3p²q
• B is wrong: This keeps the original exponents unchanged instead of subtracting them during division.
• C is wrong: This multiplies instead of divides—it adds exponents (3+2=5 total) rather than subtracting them.
• D is wrong: This doesn't actually simplify the fraction; it just rewrites it with different exponents but still in fraction form.
Simplify: $\frac{-4x^4y}{2x^2y^2}$.
Why A is correct:
Divide the coefficients: $-4 ÷ 2 = -2$. Divide the $x$ terms: $x^4 ÷ x^2 = x^2$. Divide the $y$ terms: $y ÷ y^2 = y^{-1} = 1/y$. This gives $-2x^2/y$.
Why the others are wrong:
- B ($-2x^2y$): Incorrectly kept $y$ in the numerator instead of moving it to the denominator.
- C ($-2x^6/y^2$): Added the exponents of $x$ instead of subtracting them ($x^4 ÷ x^2$ means subtract: $4-2=2$, not add).
- D ($2x^2/y$): Lost the negative sign from the original numerator.
Expand and simplify: $(3x - 2)(x + 5)$.
Why A is correct:
Use FOIL: (3x)(x) = 3x², (3x)(5) = 15x, (−2)(x) = −2x, (−2)(5) = −10. Combining: 3x² + 15x − 2x − 10 = 3x² + 13x − 10. ✓
Why the others are wrong:
- B: Forgot to combine the x terms (15x and −2x). Left them separate instead of adding them to get 13x.
- C: Incorrectly combined terms—this doesn't follow from FOIL at all.
- D: Got the constant right (−10) but only counted one of the x terms, missing the combining step.
List the factors of the term $-12m^2n$.
• Option A is correct because factors are the individual parts that multiply together to make a term. Here: $-12 \times m^2 \times n = -12m^2n$ ✓
• Option B is wrong — it's missing the negative sign. The term is negative, so $-12$ (not $12$) must be a factor.
• Option C is wrong — it has $m$ instead of $m^2$ and $n^2$ instead of $n$. These don't multiply to give the original term.
• Option D is wrong — it lists $-12m$ and $m$ separately, but $-12m$ isn't a single factor of the term; the factors must be completely simplified parts.
Which pair are like terms?
Why A is correct:
Like terms have identical variable parts with the same exponents. Both $5x^2y$ and $-3x^2y$ have $x^2y$, so only the coefficients (5 and -3) differ—they can be combined.
Why the others are wrong:
- B: $5xy^2$ has $y^2$ while $5x^2y$ has $y^1$—the exponents on $y$ don't match.
- C: $2x^3$ has exponent 3, but $2x^2$ has exponent 2—different exponents mean different terms.
- D: $4ab$ has $a^1b$ while $4a^2b$ has $a^2b$—the exponent on $a$ is different.
Simplify: $-3\bigl[4a - 2(3a - 5)\bigr]$.
• Work inside out: Start with the innermost parentheses: $2(3a - 5) = 6a - 10$.
• Substitute back: $-3[4a - (6a - 10)] = -3[4a - 6a + 10] = -3[-2a + 10]$.
• Distribute the −3: $-3 \times (-2a) + (-3) \times 10 = 6a - 30$. ✓
Why others are wrong:
- B, C, D: All have incorrect coefficients on $a$ or the constant (or both), usually from sign errors when distributing or combining like terms.
Simplify: $4x\bigl[2 - (x + 3)\bigr]$.
Correct answer: A. $-4x^2 -4x$
Work inside the brackets first: $2 - (x + 3) = 2 - x - 3 = -x - 1$.
Then distribute: $4x(-x - 1) = -4x^2 - 4x$.
Why the others are wrong:
- B: Forgot to distribute the negative sign; got $2 - x$ instead of $-x - 1$.
- C: Made a sign error in simplifying the brackets (treated it as $2 + (x + 3)$).
- D: Completely wrong signs; this would come from $4x(x + 3)$ without the subtraction.
Simplify: $3\bigl[2(x+1)-(x-2)\bigr]$.
• Start inside the brackets: $2(x+1)-(x-2) = 2x+2-x+2 = x+4$
• Multiply by 3: $3(x+4) = 3x+12$ ✓
Why others are wrong:
- B ($x+3$): You'd get this if you forgot to distribute the 3 or made an arithmetic error inside the brackets.
- C ($5x+3$): This happens if you incorrectly combine terms (like treating $2x$ and $-x$ wrong or miscalculating the constant).
- D ($3x-3$): This results from errors in distributing the negative sign or combining the constants inside the brackets.
Simplify: $7y -2\bigl[3y-(y+4)\bigr]$.
• Start inside the brackets: $3y-(y+4) = 3y-y-4 = 2y-4$
• Multiply by -2: $-2(2y-4) = -4y+8$
• Combine with $7y$: $7y-4y+8 = 3y+8$ ✓
Why others are wrong:
- B ($y+8$): Makes an arithmetic error when combining like terms; should get $3y$, not $y$
- C ($7y-2y-8$): Stops too early without fully distributing the $-2$ and combining like terms
- D ($13y-8$): Adds $7y$ and $4y$ instead of subtracting; sign error from incorrect distribution
Expand and simplify: $(3x+2)(x-5)$.
Correct answer (A): $3x^2 -13x -10$
Use FOIL or distribution: $(3x+2)(x-5) = 3x·x + 3x·(-5) + 2·x + 2·(-5) = 3x^2 - 15x + 2x - 10 = 3x^2 - 13x - 10$. The middle terms combine: $-15x + 2x = -13x$.
Why the others are wrong:
- B ($3x^2 -15x +2$): Forgot to combine like terms; left $-15x$ and $+2x$ separate instead of adding them.
- C ($3x^2 +2x -15$): Used wrong signs or forgot the $3x$ coefficient when multiplying; lost the $-15x$ term entirely.
- D ($3x^2 -10x -10$): Incorrectly combined $-15x$ and $+2x$ (got $-10x$ instead of $-13x$).
Expand and simplify: $(x+4)(x^2-4x+1)$.
Correct Answer (A): $x^3-15x+4$
Distribute $x$ and $4$ across $(x^2-4x+1)$:
- $x(x^2-4x+1) = x^3-4x^2+x$
- $4(x^2-4x+1) = 4x^2-16x+4$
- Add them: $x^3-4x^2+x+4x^2-16x+4 = x^3-15x+4$ ✓
Why others are wrong:
- B: Forgot to combine like terms; kept $4x^2$ and $-4x^2$ separate instead of canceling them.
- C: This is actually the unsimplified version before combining like terms.
- D: Arithmetic error in combining—didn't properly add/subtract the middle terms.
Expand and simplify: $(2a - b)(a + 3b)$.
Why A is correct:
Using FOIL (or distribution), multiply each term in the first bracket by each in the second: $2a \times a = 2a^2$, $2a \times 3b = 6ab$, $-b \times a = -ab$, and $-b \times 3b = -3b^2$. Combining like terms: $6ab - ab = 5ab$, giving $2a^2 + 5ab - 3b^2$.
Why the others are wrong:
- B ($2a^2 + b^2$): Missing almost all terms—appears to only multiply the first terms and squares of the second terms incorrectly.
- C ($2a^2 + 6ab - 3b^2$): Forgot to include the $-ab$ term (from $-b \times a$); combines only some products.
- D ($2a^2 + 2ab - 3b^2$): Incorrectly calculated the middle term as $2ab$ instead of $5ab$; shows incomplete or wrong multiplication.
Expand and simplify: $(x-2)(x+5)$.
Why A is correct:
Use FOIL (First, Outer, Inner, Last): $(x)(x) + (x)(5) + (-2)(x) + (-2)(5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10$ ✓
Why the others are wrong:
- B: Only multiplied the outer terms correctly ($5x$) but made errors elsewhere; missed combining $5x$ and $-2x$.
- C: Got the wrong middle term sign; this would come from $(x-2)(x-5)$ instead.
- D: Has a positive constant ($+10$) instead of negative; the product of $-2$ and $+5$ must be $-10$.
Expand and simplify: $(2x+3)^2$.
A is correct because $(2x+3)^2 = (2x+3)(2x+3)$:
- First terms: $2x \cdot 2x = 4x^2$
- Outer + Inner: $2x \cdot 3 + 3 \cdot 2x = 6x + 6x = 12x$
- Last terms: $3 \cdot 3 = 9$
- Result: $4x^2 + 12x + 9$
B is wrong because the middle term should be $12x$, not $6x$ (you need to multiply both the outer and inner products).
C is wrong because the first term should be $(2x)^2 = 4x^2$, not $2x^2$, and the last term should be $9$, not $3$.
D is wrong because the middle term is positive ($+12x$), not negative. The minus sign only appears when you have a difference like $(2x-3)^2$.
Expand and simplify: $(y-4z)^2$.
A is correct.
Use the formula $(a-b)^2 = a^2 - 2ab + b^2$:
- $a = y$, $b = 4z$
- $(y-4z)^2 = y^2 - 2(y)(4z) + (4z)^2 = y^2 - 8yz + 16z^2$ ✓
Why others are wrong:
- B: Has $+8yz$ instead of $-8yz$—this is the $(y+4z)^2$ formula.
- C: Has $-4yz$ instead of $-8yz$—forgot to multiply by 2 in the middle term.
- D: Has $4z^2$ instead of $16z^2$—forgot to square the coefficient 4 when squaring $4z$.
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