Factoring GCF - Accuplacer Aaf

1. Which of the following is the factored form of the expression $\frac{1}{2}x^3 + \frac{3}{2}x^2 - \frac{5}{2}x$?
   Answer: $\frac{1}{2}x(x^2 + 3x - 5)$ The greatest common factor includes the GCF of the numerical coefficients and the variables. The GCF of $\frac{1}{2}, \frac{3}{2}, -\frac{5}{2}$ is $\frac{1}{2}$. The GCF of $x^3, x^2, x$ is $x$. So the GCF of the entire expression is $\frac{1}{2}x$. Factoring this out gives $\frac{1}{2}x(x^2 + 3x - 5)$.
2. Two expressions $12m^4n^2 + 18m^3n^4$ and $20m^2n^3 + 15mn^5$ are to be added, then factored. What is the GCF of the resulting sum?
   Answer: $mn^2$ When you encounter polynomial addition followed by factoring, you need to combine like terms first, then find the greatest common factor (GCF) of all terms in the result. Let's add the expressions: $12m^4n^2 + 18m^3n^4 + 20m^2n^3 + 15mn^5$. Since there are no like terms to combine, this sum remains as written. To find the GCF, examine the coefficients and variables separately. For coefficients: 12, 18, 20, and 15. Find their GCF by listing factors or using prime factorization. The GCF of these numbers is 1 (since 20 and 15 share no common factors with 12 and 18 beyond 1). For variables, find the lowest power of each variable that appears in every term: - Powers of $m$: $m^4, m^3, m^2, m^1$. The lowest is $m^1$ - Powers of $n$: $n^2, n^4, n^3, n^5$. The lowest is $n^2$ Therefore, the GCF is $1 \cdot m^1 \cdot n^2 = mn^2$, which is answer B. Now for the wrong answers: A) $3mn^2$ assumes the coefficient GCF is 3, but 20 and 15 don't divide evenly by 3. C) $m^2n^2$ incorrectly uses $m^2$ when the last term only has $m^1$. D) $6mn^2$ assumes the coefficient GCF is 6, but again, this doesn't divide evenly into all four coefficients. Remember: when finding the GCF of polynomials, always use the lowest power of each variable that appears in every single term, and find the true GCF of all numerical coefficients.
3. When $12x^3y - 18x^2y^2 + 8xy^3$ is completely factored, the resulting expression is a monomial multiplied by a polynomial. What is the sum of the coefficients of the terms in the polynomial factor?
   Answer: 1 The GCF of the terms is $2xy$. Factoring this out gives $2xy(6x^2 - 9xy + 4y^2)$. The polynomial factor is $6x^2 - 9xy + 4y^2$. The coefficients of its terms are 6, -9, and 4. The sum of the coefficients is $6 + (-9) + 4 = 1$.
4. After factoring out the GCF from $15a^4b^2 - 25a^3b^4 + 10a^2b^3$, the remaining expression inside the parentheses is:
   Answer: $3a^2 - 5ab^2 + 2b$ First find the GCF: coefficients (15, 25, 10) have GCF = 5; a terms (a⁴, a³, a²) have GCF = a²; b terms (b², b⁴, b³) have GCF = b². So GCF = 5a²b². Dividing each term: 15a⁴b²÷5a²b² = 3a², 25a³b⁴÷5a²b² = 5ab², 10a²b³÷5a²b² = 2b. Choice B doesn't divide the b terms correctly. Choice C doesn't divide coefficients. Choice D doesn't divide at all.
5. The expression $7x(x-4) + (4-x)$ can be factored by first rewriting one of the binomials. Which of the following is a factor of the expression?
   Answer: $7x-1$ First, rewrite $(4-x)$ as $-1(x-4)$. The expression becomes $7x(x-4) - 1(x-4)$. Now, the common binomial factor is $(x-4)$. Factoring this out gives $(x-4)(7x-1)$. The factors are $(x-4)$ and $(7x-1)$.
6. The expression $45r^6s^3t^2 - 30r^4s^5t + 75r^5s^2t^4$ can be factored as $15r^4s^2t \cdot Q$. What is $Q$?
   Answer: $3r^2st - 2s^3 + 5rt^3$ When you see a polynomial expression that needs to be factored with a given factor, you're working with the distributive property in reverse. The key is to divide each term of the original expression by the given factor to find what remains. To find $Q$, you need to divide each term of $45r^6s^3t^2 - 30r^4s^5t + 75r^5s^2t^4$ by $15r^4s^2t$: First term: $\frac{45r^6s^3t^2}{15r^4s^2t} = 3r^2st$ Second term: $\frac{-30r^4s^5t}{15r^4s^2t} = -2s^3$ Third term: $\frac{75r^5s^2t^4}{15r^4s^2t} = 5rt^3$ Therefore, $Q = 3r^2st - 2s^3 + 5rt^3$, which is choice D. Choice A contains $9r^2st$ instead of $3r^2st$ in the first term, suggesting an error in dividing coefficients (45 ÷ 15 = 3, not 9). Choice B has the correct first and third terms but incorrectly shows $-2s^3t$ instead of $-2s^3$ in the middle term—this represents keeping an extra factor of $t$ that should have been divided out. Choice C shows $45r^2st$ and $75rt^3$, indicating the coefficients weren't divided at all, and the middle term $-30s^3$ is missing the $t$ factor entirely. When factoring polynomials on the Accuplacer, always verify your answer by multiplying the factors back together. This catches division errors with coefficients or variables and ensures you haven't missed any terms in the factorization process.
7. After factoring $42a^3b^4c^2 + 28a^2b^2c^5 - 35a^4b^3c^3$, Maria gets $7a^2b^2c^2(6ab^2 + 4c^3 - 5a^2bc)$. To verify her answer, she should:
   Answer: Multiply $7a^2b^2c^2$ by each term and check if it equals the original expression When you need to verify a factoring result, you're essentially checking whether the factored form, when expanded, gives you back the original expression. Since factoring breaks an expression into factors that multiply together, verification requires reversing this process through multiplication. To verify Maria's factoring of $42a^3b^4c^2 + 28a^2b^2c^5 - 35a^4b^3c^3 = 7a^2b^2c^2(6ab^2 + 4c^3 - 5a^2bc)$, you need to distribute $7a^2b^2c^2$ to each term in the parentheses. This means: $7a^2b^2c^2 \cdot 6ab^2 + 7a^2b^2c^2 \cdot 4c^3 + 7a^2b^2c^2 \cdot (-5a^2bc)$. When simplified, this should equal the original expression, confirming the factoring is correct. This makes B the right approach. Option A suggests adding the common factor to each term, which makes no mathematical sense for verification. Addition won't reconstruct the original expression from factored form. Option C proposes dividing the original by the common factor. While this would show you what remains after factoring out $7a^2b^2c^2$, it doesn't verify that the complete factored form is correct—you'd still need to check if the quotient matches what's in parentheses. Option D involves substituting specific values. While this might work for particular values, it doesn't prove the factoring is algebraically correct for all possible values of the variables. Remember: factoring verification always requires expanding the factored form back to the original through multiplication. Think "reverse the process" when checking factoring work.
8. Which expression represents the complete factorization of $36m^5n^3 + 48m^4n^2 - 60m^3n^4$ with the GCF factored out?
   Answer: $12m^3n^2(3m^2n + 4m - 5n^2)$ GCF of coefficients (36, 48, 60) = 12. GCF of m terms (m⁵, m⁴, m³) = m³. GCF of n terms (n³, n², n⁴) = n². So GCF = 12m³n². Factoring: 12m³n²(3m²n + 4m - 5n²). Choice B uses 6 instead of 12, making the remaining terms twice as large. Choice C incorrectly factors the exponents. Choice D uses 4 instead of 12, making remaining coefficients incorrect.
9. A student factors $21x^3y^4 - 14x^2y^2 + 35xy^5$ as $7xy^2(3x^2y^2 - 2x + 5y^3)$. What error did the student make?
   Answer: The student factored correctly with no errors made Check the factorization: GCF of (21, 14, 35) = 7. GCF of x terms (x³, x², x) = x. GCF of y terms (y⁴, y², y⁵) = y². So GCF = 7xy². Verify: 7xy²(3x²y² - 2x + 5y³) = 21x³y⁴ - 14x²y² + 35xy⁵ ✓. The factorization is correct. Choice A is wrong because 7 is the correct GCF of coefficients. Choice B is wrong because x¹ is correct. Choice D is wrong because y² is the lowest power.
10. If the greatest common factor is factored out of the expression $28a^5b^2 - 42a^3b^4$, what is the other factor?
    Answer: $2a^2 - 3b^2$ The GCF of 28 and 42 is 14. The GCF of $a^5$ and $a^3$ is $a^3$. The GCF of $b^2$ and $b^4$ is $b^2$. So the GCF is $14a^3b^2$. To find the other factor, divide each term by the GCF: $\frac{28a^5b^2}{14a^3b^2} - \frac{42a^3b^4}{14a^3b^2} = 2a^{5-3}b^{2-2} - 3a^{3-3}b^{4-2} = 2a^2 - 3b^2$.
11. When the expression $3x^3 - 75x$ is factored completely, which of the following is one of its factors?
    Answer: $x - 5$ First, factor out the GCF, which is $3x$. This gives $3x(x^2 - 25)$. The term $x^2 - 25$ is a difference of squares, which factors into $(x-5)(x+5)$. The completely factored form is $3x(x-5)(x+5)$. The factors are $3x$, $x-5$, and $x+5$.
12. Let the expression $30x^5 - 45x^3$ be factored as $P(x) \cdot Q(x)$, where $P(x)$ is the greatest common monomial factor. What is the value of $Q(1)$?
    Answer: -1 First, find the GCF, $P(x)$. The GCF of 30 and 45 is 15. The GCF of $x^5$ and $x^3$ is $x^3$. So, $P(x) = 15x^3$. Next, find $Q(x)$ by dividing the original expression by $P(x)$: $Q(x) = \frac{30x^5 - 45x^3}{15x^3} = 2x^2 - 3$. Finally, evaluate $Q(1)$: $Q(1) = 2(1)^2 - 3 = 2 - 3 = -1$.
13. If $10x^3 - 5x^2 = 0$, and the expression on the left is factored using its greatest common factor, which of the following includes all possible values of $x$?
    Answer: $0, 1/2$ First, factor the expression $10x^3 - 5x^2$. The GCF is $5x^2$. This gives $5x^2(2x - 1) = 0$. By the Zero Product Property, set each factor equal to zero. $5x^2 = 0$ gives $x=0$. And $2x - 1 = 0$ gives $2x=1$, so $x=1/2$. The solutions are 0 and 1/2.
14. What is the greatest common factor of the terms in the expression $48x^2y^3z - 36x^4y^2 + 60x^3y^4z^2$?
    Answer: $12x^2y^2$ The GCF of the coefficients 48, 36, and 60 is 12. The lowest power of $x$ common to all terms is $x^2$. The lowest power of $y$ common to all terms is $y^2$. The variable $z$ is not in the second term, so it is not a common factor. Therefore, the GCF is $12x^2y^2$.
15. The expression $-9a^4b^2 - 12a^3b^3$ is factored completely, and the greatest common factor is written with a negative coefficient. Which of the following is the other factor?
    Answer: $3a+4b$ The GCF of 9 and 12 is 3. The GCF of $a^4$ and $a^3$ is $a^3$. The GCF of $b^2$ and $b^3$ is $b^2$. The problem specifies the GCF has a negative coefficient, so the GCF is $-3a^3b^2$. Dividing the original expression by the GCF: $\frac{-9a^4b^2}{-3a^3b^2} + \frac{-12a^3b^3}{-3a^3b^2} = 3a + 4b$.
16. The expression $15x^3y - 10x^2y^2$ is factored as $A(B-C)$, where $A$ is the greatest common monomial factor. What is the value of $B-C$ when $x=2$ and $y=-1$?
    Answer: 8 First, factor the expression. The GCF of $15x^3y$ and $10x^2y^2$ is $5x^2y$. So, $A = 5x^2y$. The remaining factor is $(\frac{15x^3y}{5x^2y} - \frac{10x^2y^2}{5x^2y}) = (3x - 2y)$. This is the factor $(B-C)$. Now, evaluate $3x - 2y$ for $x=2$ and $y=-1$: $3(2) - 2(-1) = 6 - (-2) = 6 + 2 = 8$.
17. Which of the following is the complete factorization of $14x^2 + 21y^2 - 35$?
    Answer: $7(2x^2 + 3y^2 - 5)$ To factor the expression, find the GCF of the coefficients 14, 21, and 35. The GCF is 7. There are no common variable factors. Factoring out 7 gives $7(\frac{14x^2}{7} + \frac{21y^2}{7} - \frac{35}{7}) = 7(2x^2 + 3y^2 - 5)$.
18. When factoring $-8p^4q^2 + 12p^3q^4 - 16p^2q^3$, what is the factored form with the GCF (including the negative sign) completely factored out?
    Answer: $-4p^2q^2(2p^2 - 3pq^2 + 4q)$ When you see a polynomial factoring problem asking for the GCF to be completely factored out, you need to find the largest factor common to all terms, including any negative signs. Let's work through this systematically. First, examine the coefficients: -8, 12, and -16. The GCF of 8, 12, and 16 is 4. Since we have more negative terms than positive, we can factor out -4. Next, look at the variable parts: $p^4$, $p^3$, and $p^2$ share $p^2$ as their GCF. Similarly, $q^2$, $q^4$, and $q^3$ share $q^2$. So our complete GCF is $-4p^2q^2$. Now divide each term by $-4p^2q^2$: - $-8p^4q^2 ÷ (-4p^2q^2) = 2p^2$ - $12p^3q^4 ÷ (-4p^2q^2) = -3pq^2$ - $-16p^2q^3 ÷ (-4p^2q^2) = 4q$ This gives us $-4p^2q^2(2p^2 - 3pq^2 + 4q)$, which is answer C. Looking at the wrong answers: A factors out only $-2p^2q^2$, which isn't the complete GCF. B has the right remaining polynomial but uses $4p^2q^2$ instead of $-4p^2q^2$, missing the negative sign requirement. D factors out $-8p^2q^2$, which is too large and creates fractions in the remaining expression. Remember: when factoring out the GCF completely, check that no common factors remain in the resulting polynomial, and pay attention to whether factoring out a negative leads to a cleaner expression.
19. If an expression is factored as $4ab^2(3a^2 - 5b + 2a)$, which of the following was the original expression?
    Answer: $12a^3b^2 - 20ab^3 + 8a^2b^2$ To find the original expression, distribute the monomial factor $4ab^2$ to each term inside the parentheses: $4ab^2(3a^2) - 4ab^2(5b) + 4ab^2(2a) = 12a^{1+2}b^2 - 20ab^{2+1} + 8a^{1+1}b^2 = 12a^3b^2 - 20ab^3 + 8a^2b^2$.
20. Which of the following is the completely factored form of $-15m^4n^2p - 25m^3n^3 + 10m^2n^2p^2$, where the GCF has a positive numerical coefficient?
    Answer: $5m^2n^2(-3m^2p - 5mn + 2p^2)$ The GCF of 15, 25, and 10 is 5. The GCF of the $m$ terms is $m^2$. The GCF of the $n$ terms is $n^2$. The variable $p$ is not in all terms. So, the GCF is $5m^2n^2$. Dividing each term by the GCF gives: $\frac{-15m^4n^2p}{5m^2n^2} = -3m^2p$; $\frac{-25m^3n^3}{5m^2n^2} = -5mn$; $\frac{10m^2n^2p^2}{5m^2n^2} = 2p^2$. The factored form is $5m^2n^2(-3m^2p - 5mn + 2p^2)$.

Start this quiz