Distributing & Factoring GCF - Accuplacer Qas

An expression is given by $7 - (4x - 5)$ . Which of the following is an equivalent expression?
Answer: $12 - 4x$ The expression $7 - (4x - 5)$ is equivalent to $7 - 1(4x - 5)$ . Distribute the -1 to both terms inside the parentheses: $-1 \times 4x = -4x$ and $-1 \times -5 = +5$ . The expression becomes $7 - 4x + 5$ . Combine the constant terms $7 + 5 = 12$ to get the final simplified expression $12 - 4x$ .
When factoring the expression $12x^3y^2 + 18x^2y^3 - 24xy^4$ , what is the greatest common factor that can be factored out?
Answer: $6xy^2$ To find the GCF, we identify the highest power of each variable and the largest coefficient that divides all terms. For coefficients: GCD(12, 18, 24) = 6. For x: minimum power is x¹. For y: minimum power is y². Therefore, GCF = 6xy². Choice B incorrectly uses x², choice C uses coefficient 12 (doesn't divide 18 evenly), and choice D uses y³ (first term only has y²).
Which of the following expressions represents the phrase 'the product of $4x$ and the quantity $2x$ less than 7'?
Answer: $28x - 8x^2$ First, translate the phrase into an algebraic expression. 'The quantity $2x$ less than 7' translates to $7 - 2x$ . 'The product of $4x$ and that quantity' means we multiply $4x$ by $(7 - 2x)$ , which is written as $4x(7 - 2x)$ . To simplify, distribute the $4x$ : $4x \times 7 = 28x$ and $4x \times (-2x) = -8x^2$ . The resulting expression is $28x - 8x^2$ .
Two students factor the same expression differently: Student A gets $6x(2x^2 - 3x + 1)$ and Student B gets $3x(4x^2 - 6x + 2)$ . If both factorizations are mathematically correct, what can be concluded?
Answer: Student A factored out the GCF completely, but Student B did not When you encounter factoring problems where students get different results, you need to check whether each factorization is complete by examining the greatest common factor (GCF). Let's verify both factorizations by expanding them. Student A: $6x(2x^2 - 3x + 1) = 12x^3 - 18x^2 + 6x$ . Student B: $3x(4x^2 - 6x + 2) = 12x^3 - 18x^2 + 6x$ . Both give the same result, so both factorizations are mathematically correct. However, the key difference is completeness. Student A factored out $6x$ , leaving the expression $(2x^2 - 3x + 1)$ inside the parentheses. Looking at the coefficients 2, -3, and 1, their GCF is 1, so no further factoring is possible. Student B factored out only $3x$ , leaving $(4x^2 - 6x + 2)$ . The coefficients 4, -6, and 2 have a GCF of 2, meaning Student B could factor out an additional 2: $3x(4x^2 - 6x + 2) = 3x \cdot 2(2x^2 - 3x + 1) = 6x(2x^2 - 3x + 1)$ . Choice A is backwards—Student A found the complete GCF. Choice C is wrong because Student B didn't factor completely. Choice D incorrectly assumes different-looking results can't both be mathematically valid. Choice B correctly identifies that Student A factored out the complete GCF while Student B stopped partway. Remember: when factoring, always check if the terms inside parentheses share any common factors. Complete factorization means no further common factors remain.
The profit, in dollars, from selling $n$ items is given by the expression $20n - (5n + 50)$ . Which of the following is an equivalent expression for the profit?
Answer: $15n - 50$ To simplify the expression $20n - (5n + 50)$ , distribute the negative sign to both terms inside the parentheses. This changes the expression to $20n - 5n - 50$ . Then, combine the like terms: $20n - 5n = 15n$ . The simplified expression is $15n - 50$ .
The expression $24x^3y^2 + 16x^2y^3 - 40xy^4$ can be factored as $8xy^2(3x^2 + 2xy - 5y^2)$ . If instead we factor out only $4xy$ , what would be the constant term in the resulting trinomial?
Answer: $-10y^3$ When you encounter factoring problems like this, you're working with the distributive property in reverse. The key is understanding what happens when you factor out different amounts from a polynomial expression. Let's factor out $4xy$ from each term in $24x^3y^2 + 16x^2y^3 - 40xy^4$ : For the first term: $24x^3y^2 = 4xy \cdot 6x^2y$ For the second term: $16x^2y^3 = 4xy \cdot 4xy^2$ For the third term: $-40xy^4 = 4xy \cdot (-10y^3)$ This gives us: $4xy(6x^2y + 4xy^2 - 10y^3)$ The constant term in the resulting trinomial is $-10y^3$ , making C correct. Let's examine why the other choices are wrong. Choice A ( $-10y^2$ ) has the right coefficient but the wrong power of $y$ - you'd get this if you mistakenly factored out $4xy^2$ instead of $4xy$ . Choice B ( $-5y^2$ ) would result from factoring out $8xy^2$ , which is what was given in the original factorization. Choice D ( $-40y^3$ ) would mean you forgot to divide the original term by the factored amount entirely. The trap here is confusing different factorization scenarios. Always divide each original term by exactly what you're factoring out. When factoring problems give you one factorization and ask about another, work through the division step-by-step rather than trying to convert between the two factorizations directly.
Which of the following is the factored form of $x(3x + 4) - 2(3x + 4)$ ?
Answer: $(x-2)(3x+4)$ This expression can be factored by recognizing that $(3x+4)$ is a common factor. Factoring out $(3x+4)$ gives $(x-2)(3x+4)$ . To verify: expanding $(x-2)(3x+4)$ yields $3x^2 + 4x - 6x - 8 = 3x^2 - 2x - 8$ , but the factored form $(x-2)(3x+4)$ is the most simplified equivalent expression.
A student claims that $4x^2 - 8x + 12$ factors as $4(x^2 - 2x + 3)$ . What error, if any, did the student make?
Answer: No error; the factorization is completely correct The student correctly factored out the GCF of 4: 4x² - 8x + 12 = 4(x² - 2x + 3). Choice B is wrong because 4 is indeed the GCF of 4, 8, and 12. Choice C is incorrect because x² - 2x + 3 cannot factor further over the reals (discriminant = 4 - 12 = -8 < 0). Choice D is redundant with A but uses different wording that might confuse students about whether further factoring is possible.
The area of a rectangular garden is given by the expression $5(3x - 2)$ square meters. The width of the garden is 5 meters. If the expression for the area is expanded to $15x - 10$ , what does the term $15x$ represent in this context?
Answer: The area of a larger rectangle with dimensions 5 meters by $3x$ meters The area of a rectangle is length times width. Here, the width is 5 and the length is $(3x - 2)$ . The expanded area, $15x - 10$ , can be interpreted as the area of a rectangle with width 5 and length $3x$ (which is $15x$ ) from which a smaller rectangular area of $5 \times 2 = 10$ has been subtracted. Therefore, the term $15x$ represents the area of that larger, initial rectangle.
A triangle has side lengths of $(4x+8)$ , $(6x+12)$ , and $(2x+4)$ . Which of the following expressions represents the perimeter of the triangle in factored form?
Answer: $12(x+2)$ First, find the perimeter by adding the lengths of the three sides: $(4x+8) + (6x+12) + (2x+4)$ . Combine like terms: $(4x+6x+2x) + (8+12+4) = 12x + 24$ . Next, factor the expression for the perimeter. The greatest common factor of $12x$ and $24$ is 12. Factoring out 12 gives $12(x+2)$ .
Which expression is equivalent to $-15m^3n + 25m^2n^2 - 35mn^3$ after factoring out the GCF?
Answer: $-5mn(3m^2 - 5mn + 7n^2)$ When you see a polynomial expression that needs factoring, your goal is to identify the Greatest Common Factor (GCF) among all terms and factor it out completely. To find the GCF of $-15m^3n + 25m^2n^2 - 35mn^3$ , examine both the coefficients and variables separately. For coefficients: the GCF of 15, 25, and 35 is 5. For variables: each term contains at least $m^1$ and $n^1$ , so the variable part of the GCF is $mn$ . However, since the first term is negative, you can factor out $-5mn$ to avoid having a negative leading coefficient in the parentheses. Factoring out $-5mn$ : - $-15m^3n ÷ (-5mn) = 3m^2$ - $25m^2n^2 ÷ (-5mn) = -5mn$ - $-35mn^3 ÷ (-5mn) = 7n^2$ This gives us $-5mn(3m^2 - 5mn + 7n^2)$ , which is answer D. Let's check why the other options are incorrect: Choice A uses $5mn$ instead of $-5mn$ as the GCF, which would change the signs inside the parentheses incorrectly. Choice B has the right GCF but wrong signs throughout the parentheses—this happens when students make sign errors during division. Choice C incorrectly shows addition signs instead of the proper subtraction, suggesting confusion about how negative division affects the terms. Study tip: Always verify your factoring by distributing the GCF back through the parentheses. If you get the original expression, your factorization is correct.
The expression $\frac{1}{2}x^2 + \frac{3}{2}x$ can be factored. Which of the following is the completely factored form?
Answer: $\frac{1}{2}x(x + 3)$ To factor completely, find the greatest common factor (GCF). Both terms have a numerical factor of $\frac{1}{2}$ and a variable factor of $x$ . Therefore, the GCF is $\frac{1}{2}x$ . Factoring this out from the first term gives $\frac{\frac{1}{2}x^2}{\frac{1}{2}x} = x$ . Factoring it out from the second term gives $\frac{\frac{3}{2}x}{\frac{1}{2}x} = 3$ . The completely factored form is $\frac{1}{2}x(x + 3)$ .
Which of the following is equivalent to $-8a^3b - 12ab^2$ ?
Answer: $-4ab(2a^2 + 3b)$ The greatest common factor (GCF) of 8 and 12 is 4. The GCF of $a^3$ and $a$ is $a$ . The GCF of $b$ and $b^2$ is $b$ . So the GCF is $4ab$ . Since the leading term is negative, we factor out $-4ab$ . Dividing each term by $-4ab$ : $\frac{-8a^3b}{-4ab} = 2a^2$ and $\frac{-12ab^2}{-4ab} = 3b$ . This gives the factored form $-4ab(2a^2 + 3b)$ .
Which of the following is the complete factorization of $12x^4y^2 - 18x^2y^3$ ?
Answer: $6x^2y^2(2x^2 - 3y)$ To factor the expression completely, find the greatest common factor (GCF) of both terms. The GCF of the coefficients 12 and 18 is 6. The GCF of the variable parts $x^4y^2$ and $x^2y^3$ is $x^2y^2$ (the lowest power of each variable). So, the GCF of the entire expression is $6x^2y^2$ . Factoring this out gives $6x^2y^2(2x^2 - 3y)$ .
What is the greatest common factor of the terms in the expression $15x^3y^4z^2 + 25x^2y^5z$ ?
Answer: $5x^2y^4z$ To find the greatest common factor (GCF), find the GCF of the coefficients and each variable part. The GCF of 15 and 25 is 5. For the variables, take the lowest power that appears in both terms. The GCF of $x^3$ and $x^2$ is $x^2$ . The GCF of $y^4$ and $y^5$ is $y^4$ . The GCF of $z^2$ and $z$ is $z$ . Combining these gives the GCF of the entire expression: $5x^2y^4z$ .
The total cost for a group to rent a bus is $100 + 4n$ dollars, where $n$ is the number of people. The total revenue from the trip is $10n$ dollars. If profit is revenue minus cost, which expression represents the profit for the trip in factored form?
Answer: $2(3n - 50)$ Profit is calculated as Revenue - Cost. The expression for profit is $10n - (100 + 4n)$ . First, distribute the negative sign: $10n - 100 - 4n$ . Then, combine like terms: $10n - 4n = 6n$ , so the profit is $6n - 100$ . To write this in factored form, find the greatest common factor of $6n$ and 100, which is 2. Factoring out 2 gives $2(3n - 50)$ .
Which of the following is the result of distributing $-3x^2y$ to the expression $(5xy^2 - 2x^2)$ ?
Answer: $-15x^3y^3 + 6x^4y$ Apply the distributive property. Multiply $-3x^2y$ by each term in the parentheses. For the first term: $(-3x^2y)(5xy^2) = (-3)(5)(x^2 \cdot x)(y \cdot y^2) = -15x^3y^3$ . For the second term: $(-3x^2y)(-2x^2) = (-3)(-2)(x^2 \cdot x^2)(y) = 6x^4y$ . The resulting expression is $-15x^3y^3 + 6x^4y$ .
An expression is given by $3x(y-2) + 5(y-2)$ . If this expression is factored completely, which of the following is one of its factors?
Answer: $3x+5$ This expression can be factored by grouping. Notice that $(y-2)$ is a common binomial factor to both terms. Factoring out $(y-2)$ leaves $3x$ from the first term and $+5$ from the second term. The completely factored expression is $(y-2)(3x+5)$ . The two factors are $(y-2)$ and $(3x+5)$ .
The surface area of a cylinder is given by the formula $A = 2\pi r^2 + 2\pi rh$ . Which of the following shows the formula with the greatest common factor factored out?
Answer: $2\pi r(r + h)$ In the expression $2\pi r^2 + 2\pi rh$ , we look for the greatest common factor (GCF) of the two terms. Both terms share the factors 2, $\pi$ , and $r$ . Therefore, the GCF is $2\pi r$ . Factoring $2\pi r$ out of the first term, $2\pi r^2$ , leaves $r$ . Factoring it out of the second term, $2\pi rh$ , leaves $h$ . The factored form is $2\pi r(r + h)$ .
The expression $24x^3 - 36x^2$ can be written in the form $A \cdot B$ , where $A$ is the greatest common monomial factor. What is the expression for $B$ ?
Answer: $2x - 3$ First, find the greatest common factor (GCF) of $24x^3$ and $36x^2$ . The GCF of 24 and 36 is 12. The GCF of $x^3$ and $x^2$ is $x^2$ . Thus, the GCF, which is $A$ , is $12x^2$ . To find $B$ , divide the original expression by $A$ : $\frac{24x^3 - 36x^2}{12x^2} = \frac{24x^3}{12x^2} - \frac{36x^2}{12x^2} = 2x - 3$ . So, $B = 2x - 3$ .

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