Math worksheet for factoring out the greatest common factor (GCF) from algebraic expressions.
Worksheet titled "Factoring out the GCF" with ten algebraic expressions to factor, including variables and exponents, from Math Monks.
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Step-by-step solution for: Factoring Polynomials Worksheets with Answer Key
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Show Answer Key & Explanations
Step-by-step solution for: Factoring Polynomials Worksheets with Answer Key
To solve the problem of factoring out the greatest common factor (GCF) from each expression, we need to identify the largest factor that divides all terms in the expression. This includes both numerical coefficients and variable factors. Let's go through each expression step by step.
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1. Identify the GCF of the coefficients: The coefficients are \( 15, -25, \) and \( 55 \). The GCF of these numbers is \( 5 \).
2. Identify the GCF of the variables: The variables are \( x^3, x^2, \) and \( x \). The smallest power of \( x \) is \( x \). So, the GCF of the variables is \( x \).
3. Combine the GCF: The overall GCF is \( 5x \).
4. Factor out \( 5x \):
\[
15x^3 - 25x^2 + 55x = 5x(3x^2 - 5x + 11)
\]
Answer: \( 5x(3x^2 - 5x + 11) \)
---
1. Identify the GCF of the coefficients: The coefficients are \( 36, 63, \) and \( -27 \). The GCF of these numbers is \( 9 \).
2. Identify the GCF of the variables: The variables are \( x^3, x^2, \) and \( x \). The smallest power of \( x \) is \( x \). So, the GCF of the variables is \( x \).
3. Combine the GCF: The overall GCF is \( 9x \).
4. Factor out \( 9x \):
\[
36x^3 + 63x^2 - 27x = 9x(4x^2 + 7x - 3)
\]
Answer: \( 9x(4x^2 + 7x - 3) \)
---
1. Identify the GCF of the coefficients: The coefficients are \( 25 \) and \( -35 \). The GCF of these numbers is \( 5 \).
2. Identify the GCF of the variables: The variables are \( x^{12} \) and \( x^6 \). The smallest power of \( x \) is \( x^6 \). So, the GCF of the variables is \( x^6 \).
3. Combine the GCF: The overall GCF is \( 5x^6 \).
4. Factor out \( 5x^6 \):
\[
25x^{12} - 35x^6 = 5x^6(5x^6 - 7)
\]
Answer: \( 5x^6(5x^6 - 7) \)
---
1. Identify the GCF of the coefficients: The coefficients are \( -6 \) and \( -6 \). The GCF of these numbers is \( -6 \).
2. Identify the GCF of the variables: The variables are \( p^5 \) and \( p^4 \). The smallest power of \( p \) is \( p^4 \). So, the GCF of the variables is \( p^4 \).
3. Combine the GCF: The overall GCF is \( -6p^4 \).
4. Factor out \( -6p^4 \):
\[
-6p^5 - 6p^4 = -6p^4(p + 1)
\]
Answer: \( -6p^4(p + 1) \)
---
1. Identify the GCF of the coefficients: The coefficients are \( 72, -72, \) and \( -80 \). The GCF of these numbers is \( 8 \).
2. Identify the GCF of the variables: The variables are \( x^5, x^3, \) and \( x^2 \). The smallest power of \( x \) is \( x^2 \). So, the GCF of the variables is \( x^2 \).
3. Combine the GCF: The overall GCF is \( 8x^2 \).
4. Factor out \( 8x^2 \):
\[
72x^5 - 72x^3 - 80x^2 = 8x^2(9x^3 - 9x - 10)
\]
Answer: \( 8x^2(9x^3 - 9x - 10) \)
---
1. Identify the GCF of the coefficients: The coefficients are \( 6, 2, \) and \( -4 \). The GCF of these numbers is \( 2 \).
2. Identify the GCF of the variables: The variables are \( x^2yz, xy^2z, \) and \( xyz \). The smallest powers are \( x, y, \) and \( z \). So, the GCF of the variables is \( xyz \).
3. Combine the GCF: The overall GCF is \( 2xyz \).
4. Factor out \( 2xyz \):
\[
6x^2yz + 2xy^2z - 4xyz = 2xyz(3x + y - 2)
\]
Answer: \( 2xyz(3x + y - 2) \)
---
1. Identify the GCF of the coefficients: The coefficients are \( -16, 24, \) and \( -32 \). The GCF of these numbers is \( -8 \).
2. Identify the GCF of the variables: The variables are \( p^3q^2, p^2q^3, \) and \( p^4q \). The smallest powers are \( p^2 \) and \( q \). So, the GCF of the variables is \( p^2q \).
3. Combine the GCF: The overall GCF is \( -8p^2q \).
4. Factor out \( -8p^2q \):
\[
-16p^3q^2 + 24p^2q^3 - 32p^4q = -8p^2q(2pq - 3q^2 + 4p^2)
\]
Answer: \( -8p^2q(2pq - 3q^2 + 4p^2) \)
---
1. Simplify the second term: Notice that \( 9 - a = -(a - 9) \). So, the expression becomes:
\[
7wx(a - 9) - 10w(-(a - 9)) = 7wx(a - 9) + 10w(a - 9)
\]
2. Identify the GCF: The GCF of the terms is \( w(a - 9) \).
3. Factor out \( w(a - 9) \):
\[
7wx(a - 9) + 10w(a - 9) = w(a - 9)(7x + 10)
\]
Answer: \( w(a - 9)(7x + 10) \)
---
1. Identify the GCF of the coefficients: The coefficients are \( 36, 40, -160, \) and \( -20 \). The GCF of these numbers is \( 4 \).
2. Identify the GCF of the variables: The variables are \( w^5, w^4, w^3, \) and \( w^2 \). The smallest power of \( w \) is \( w^2 \). So, the GCF of the variables is \( w^2 \).
3. Combine the GCF: The overall GCF is \( 4w^2 \).
4. Factor out \( 4w^2 \):
\[
36w^5 + 40w^4 - 160w^3 - 20w^2 = 4w^2(9w^3 + 10w^2 - 40w - 5)
\]
Answer: \( 4w^2(9w^3 + 10w^2 - 40w - 5) \)
---
1. Simplify the second term: Notice that \( 10 + m = m + 10 \). So, the expression becomes:
\[
bc(m + 10) - 5b(m + 10)
\]
2. Identify the GCF: The GCF of the terms is \( b(m + 10) \).
3. Factor out \( b(m + 10) \):
\[
bc(m + 10) - 5b(m + 10) = b(m + 10)(c - 5)
\]
Answer: \( b(m + 10)(c - 5) \)
---
\[
\boxed{
\begin{aligned}
1. & \ 5x(3x^2 - 5x + 11) \\
2. & \ 9x(4x^2 + 7x - 3) \\
3. & \ 5x^6(5x^6 - 7) \\
4. & \ -6p^4(p + 1) \\
5. & \ 8x^2(9x^3 - 9x - 10) \\
6. & \ 2xyz(3x + y - 2) \\
7. & \ -8p^2q(2pq - 3q^2 + 4p^2) \\
8. & \ w(a - 9)(7x + 10) \\
9. & \ 4w^2(9w^3 + 10w^2 - 40w - 5) \\
10. & \ b(m + 10)(c - 5)
\end{aligned}
}
\]
---
Expression 1: \( 15x^3 - 25x^2 + 55x \)
1. Identify the GCF of the coefficients: The coefficients are \( 15, -25, \) and \( 55 \). The GCF of these numbers is \( 5 \).
2. Identify the GCF of the variables: The variables are \( x^3, x^2, \) and \( x \). The smallest power of \( x \) is \( x \). So, the GCF of the variables is \( x \).
3. Combine the GCF: The overall GCF is \( 5x \).
4. Factor out \( 5x \):
\[
15x^3 - 25x^2 + 55x = 5x(3x^2 - 5x + 11)
\]
Answer: \( 5x(3x^2 - 5x + 11) \)
---
Expression 2: \( 36x^3 + 63x^2 - 27x \)
1. Identify the GCF of the coefficients: The coefficients are \( 36, 63, \) and \( -27 \). The GCF of these numbers is \( 9 \).
2. Identify the GCF of the variables: The variables are \( x^3, x^2, \) and \( x \). The smallest power of \( x \) is \( x \). So, the GCF of the variables is \( x \).
3. Combine the GCF: The overall GCF is \( 9x \).
4. Factor out \( 9x \):
\[
36x^3 + 63x^2 - 27x = 9x(4x^2 + 7x - 3)
\]
Answer: \( 9x(4x^2 + 7x - 3) \)
---
Expression 3: \( 25x^{12} - 35x^6 \)
1. Identify the GCF of the coefficients: The coefficients are \( 25 \) and \( -35 \). The GCF of these numbers is \( 5 \).
2. Identify the GCF of the variables: The variables are \( x^{12} \) and \( x^6 \). The smallest power of \( x \) is \( x^6 \). So, the GCF of the variables is \( x^6 \).
3. Combine the GCF: The overall GCF is \( 5x^6 \).
4. Factor out \( 5x^6 \):
\[
25x^{12} - 35x^6 = 5x^6(5x^6 - 7)
\]
Answer: \( 5x^6(5x^6 - 7) \)
---
Expression 4: \( -6p^5 - 6p^4 \)
1. Identify the GCF of the coefficients: The coefficients are \( -6 \) and \( -6 \). The GCF of these numbers is \( -6 \).
2. Identify the GCF of the variables: The variables are \( p^5 \) and \( p^4 \). The smallest power of \( p \) is \( p^4 \). So, the GCF of the variables is \( p^4 \).
3. Combine the GCF: The overall GCF is \( -6p^4 \).
4. Factor out \( -6p^4 \):
\[
-6p^5 - 6p^4 = -6p^4(p + 1)
\]
Answer: \( -6p^4(p + 1) \)
---
Expression 5: \( 72x^5 - 72x^3 - 80x^2 \)
1. Identify the GCF of the coefficients: The coefficients are \( 72, -72, \) and \( -80 \). The GCF of these numbers is \( 8 \).
2. Identify the GCF of the variables: The variables are \( x^5, x^3, \) and \( x^2 \). The smallest power of \( x \) is \( x^2 \). So, the GCF of the variables is \( x^2 \).
3. Combine the GCF: The overall GCF is \( 8x^2 \).
4. Factor out \( 8x^2 \):
\[
72x^5 - 72x^3 - 80x^2 = 8x^2(9x^3 - 9x - 10)
\]
Answer: \( 8x^2(9x^3 - 9x - 10) \)
---
Expression 6: \( 6x^2yz + 2xy^2z - 4xyz \)
1. Identify the GCF of the coefficients: The coefficients are \( 6, 2, \) and \( -4 \). The GCF of these numbers is \( 2 \).
2. Identify the GCF of the variables: The variables are \( x^2yz, xy^2z, \) and \( xyz \). The smallest powers are \( x, y, \) and \( z \). So, the GCF of the variables is \( xyz \).
3. Combine the GCF: The overall GCF is \( 2xyz \).
4. Factor out \( 2xyz \):
\[
6x^2yz + 2xy^2z - 4xyz = 2xyz(3x + y - 2)
\]
Answer: \( 2xyz(3x + y - 2) \)
---
Expression 7: \( -16p^3q^2 + 24p^2q^3 - 32p^4q \)
1. Identify the GCF of the coefficients: The coefficients are \( -16, 24, \) and \( -32 \). The GCF of these numbers is \( -8 \).
2. Identify the GCF of the variables: The variables are \( p^3q^2, p^2q^3, \) and \( p^4q \). The smallest powers are \( p^2 \) and \( q \). So, the GCF of the variables is \( p^2q \).
3. Combine the GCF: The overall GCF is \( -8p^2q \).
4. Factor out \( -8p^2q \):
\[
-16p^3q^2 + 24p^2q^3 - 32p^4q = -8p^2q(2pq - 3q^2 + 4p^2)
\]
Answer: \( -8p^2q(2pq - 3q^2 + 4p^2) \)
---
Expression 8: \( 7wx(a - 9) - 10w(9 - a) \)
1. Simplify the second term: Notice that \( 9 - a = -(a - 9) \). So, the expression becomes:
\[
7wx(a - 9) - 10w(-(a - 9)) = 7wx(a - 9) + 10w(a - 9)
\]
2. Identify the GCF: The GCF of the terms is \( w(a - 9) \).
3. Factor out \( w(a - 9) \):
\[
7wx(a - 9) + 10w(a - 9) = w(a - 9)(7x + 10)
\]
Answer: \( w(a - 9)(7x + 10) \)
---
Expression 9: \( 36w^5 + 40w^4 - 160w^3 - 20w^2 \)
1. Identify the GCF of the coefficients: The coefficients are \( 36, 40, -160, \) and \( -20 \). The GCF of these numbers is \( 4 \).
2. Identify the GCF of the variables: The variables are \( w^5, w^4, w^3, \) and \( w^2 \). The smallest power of \( w \) is \( w^2 \). So, the GCF of the variables is \( w^2 \).
3. Combine the GCF: The overall GCF is \( 4w^2 \).
4. Factor out \( 4w^2 \):
\[
36w^5 + 40w^4 - 160w^3 - 20w^2 = 4w^2(9w^3 + 10w^2 - 40w - 5)
\]
Answer: \( 4w^2(9w^3 + 10w^2 - 40w - 5) \)
---
Expression 10: \( bc(m + 10) - 5b(10 + m) \)
1. Simplify the second term: Notice that \( 10 + m = m + 10 \). So, the expression becomes:
\[
bc(m + 10) - 5b(m + 10)
\]
2. Identify the GCF: The GCF of the terms is \( b(m + 10) \).
3. Factor out \( b(m + 10) \):
\[
bc(m + 10) - 5b(m + 10) = b(m + 10)(c - 5)
\]
Answer: \( b(m + 10)(c - 5) \)
---
Final Answers
\[
\boxed{
\begin{aligned}
1. & \ 5x(3x^2 - 5x + 11) \\
2. & \ 9x(4x^2 + 7x - 3) \\
3. & \ 5x^6(5x^6 - 7) \\
4. & \ -6p^4(p + 1) \\
5. & \ 8x^2(9x^3 - 9x - 10) \\
6. & \ 2xyz(3x + y - 2) \\
7. & \ -8p^2q(2pq - 3q^2 + 4p^2) \\
8. & \ w(a - 9)(7x + 10) \\
9. & \ 4w^2(9w^3 + 10w^2 - 40w - 5) \\
10. & \ b(m + 10)(c - 5)
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of gcf monomials worksheet.
