Math worksheet for factoring polynomials using GCF and Zero Product Property.

To solve the given problems, we need to factor each polynomial by finding the Greatest Common Factor (GCF) of its terms. Let's go through each problem step by step.

Problem 1: \( 32b + 8 \)

1. Identify the GCF of the coefficients \( 32 \) and \( 8 \):
- The factors of \( 32 \) are \( 1, 2, 4, 8, 16, 32 \).
- The factors of \( 8 \) are \( 1, 2, 4, 8 \).
- The greatest common factor is \( 8 \).

2. Factor out the GCF:
\[
32b + 8 = 8(4b + 1)
\]

Problem 2: \( -5n + 15 \)

1. Identify the GCF of the coefficients \( -5 \) and \( 15 \):
- The factors of \( -5 \) are \( -5, -1, 1, 5 \).
- The factors of \( 15 \) are \( 1, 3, 5, 15 \).
- The greatest common factor is \( 5 \).

2. Factor out the GCF:
\[
-5n + 15 = 5(-n + 3)
\]

Problem 3: \( 35a^4 + 30a^2 \)

1. Identify the GCF of the coefficients \( 35 \) and \( 30 \):
- The factors of \( 35 \) are \( 1, 5, 7, 35 \).
- The factors of \( 30 \) are \( 1, 2, 3, 5, 6, 10, 15, 30 \).
- The greatest common factor is \( 5 \).

2. Identify the GCF of the variables \( a^4 \) and \( a^2 \):
- The lowest power of \( a \) is \( a^2 \).

3. Factor out the GCF:
\[
35a^4 + 30a^2 = 5a^2(7a^2 + 6)
\]

Problem 4: \( 6n^3 + 48n^2 \)

1. Identify the GCF of the coefficients \( 6 \) and \( 48 \):
- The factors of \( 6 \) are \( 1, 2, 3, 6 \).
- The factors of \( 48 \) are \( 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 \).
- The greatest common factor is \( 6 \).

2. Identify the GCF of the variables \( n^3 \) and \( n^2 \):
- The lowest power of \( n \) is \( n^2 \).

3. Factor out the GCF:
\[
6n^3 + 48n^2 = 6n^2(n + 8)
\]

Problem 5: \( 27x^4y + 45x^2 \)

1. Identify the GCF of the coefficients \( 27 \) and \( 45 \):
- The factors of \( 27 \) are \( 1, 3, 9, 27 \).
- The factors of \( 45 \) are \( 1, 3, 5, 9, 15, 45 \).
- The greatest common factor is \( 9 \).

2. Identify the GCF of the variables \( x^4y \) and \( x^2 \):
- The lowest power of \( x \) is \( x^2 \).
- The variable \( y \) is not present in the second term, so it is not part of the GCF.

3. Factor out the GCF:
\[
27x^4y + 45x^2 = 9x^2(3x^2y + 5)
\]

Problem 6: \( 8ab^4 + 32ab^3 \)

1. Identify the GCF of the coefficients \( 8 \) and \( 32 \):
- The factors of \( 8 \) are \( 1, 2, 4, 8 \).
- The factors of \( 32 \) are \( 1, 2, 4, 8, 16, 32 \).
- The greatest common factor is \( 8 \).

2. Identify the GCF of the variables \( ab^4 \) and \( ab^3 \):
- The lowest power of \( a \) is \( a \).
- The lowest power of \( b \) is \( b^3 \).

3. Factor out the GCF:
\[
8ab^4 + 32ab^3 = 8ab^3(b + 4)
\]

Problem 7: \( 30x^2 + 24x + 21 \)

1. Identify the GCF of the coefficients \( 30 \), \( 24 \), and \( 21 \):
- The factors of \( 30 \) are \( 1, 2, 3, 5, 6, 10, 15, 30 \).
- The factors of \( 24 \) are \( 1, 2, 3, 4, 6, 8, 12, 24 \).
- The factors of \( 21 \) are \( 1, 3, 7, 21 \).
- The greatest common factor is \( 3 \).

2. Factor out the GCF:
\[
30x^2 + 24x + 21 = 3(10x^2 + 8x + 7)
\]

Problem 8: \( 21k^4 + 21k^2 - 7k \)

1. Identify the GCF of the coefficients \( 21 \), \( 21 \), and \( -7 \):
- The factors of \( 21 \) are \( 1, 3, 7, 21 \).
- The factors of \( -7 \) are \( -7, -1, 1, 7 \).
- The greatest common factor is \( 7 \).

2. Identify the GCF of the variables \( k^4 \), \( k^2 \), and \( k \):
- The lowest power of \( k \) is \( k \).

3. Factor out the GCF:
\[
21k^4 + 21k^2 - 7k = 7k(3k^3 + 3k - 1)
\]

Problem 9: \( 72n^3 + 8n^2 + 40n - 56 \)

1. Identify the GCF of the coefficients \( 72 \), \( 8 \), \( 40 \), and \( -56 \):
- The factors of \( 72 \) are \( 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 \).
- The factors of \( 8 \) are \( 1, 2, 4, 8 \).
- The factors of \( 40 \) are \( 1, 2, 4, 5, 8, 10, 20, 40 \).
- The factors of \( -56 \) are \( -56, -28, -14, -8, -7, -4, -2, -1, 1, 2, 4, 7, 8, 14, 28, 56 \).
- The greatest common factor is \( 8 \).

2. Factor out the GCF:
\[
72n^3 + 8n^2 + 40n - 56 = 8(9n^3 + n^2 + 5n - 7)
\]

Problem 10: \( 18m^3 - 10m^4 + 12m^5 + 12m^9 \)

1. Identify the GCF of the coefficients \( 18 \), \( -10 \), \( 12 \), and \( 12 \):
- The factors of \( 18 \) are \( 1, 2, 3, 6, 9, 18 \).
- The factors of \( -10 \) are \( -10, -5, -2, -1, 1, 2, 5, 10 \).
- The factors of \( 12 \) are \( 1, 2, 3, 4, 6, 12 \).
- The greatest common factor is \( 2 \).

2. Identify the GCF of the variables \( m^3 \), \( m^4 \), \( m^5 \), and \( m^9 \):
- The lowest power of \( m \) is \( m^3 \).

3. Factor out the GCF:
\[
18m^3 - 10m^4 + 12m^5 + 12m^9 = 2m^3(9 - 5m + 6m^2 + 6m^6)
\]

Problem 11: \( 40uv^3 + 100u^2 + 70uv \)

1. Identify the GCF of the coefficients \( 40 \), \( 100 \), and \( 70 \):
- The factors of \( 40 \) are \( 1, 2, 4, 5, 8, 10, 20, 40 \).
- The factors of \( 100 \) are \( 1, 2, 4, 5, 10, 20, 25, 50, 100 \).
- The factors of \( 70 \) are \( 1, 2, 5, 7, 10, 14, 35, 70 \).
- The greatest common factor is \( 10 \).

2. Identify the GCF of the variables \( uv^3 \), \( u^2 \), and \( uv \):
- The lowest power of \( u \) is \( u \).
- The lowest power of \( v \) is \( v \).

3. Factor out the GCF:
\[
40uv^3 + 100u^2 + 70uv = 10u(4v^3 + 10u + 7v)
\]

Problem 12: \( 2u^6v - 9u^2v - 2uv^2 + 6uv \)

1. Identify the GCF of the coefficients \( 2 \), \( -9 \), \( -2 \), and \( 6 \):
- The factors of \( 2 \) are \( 1, 2 \).
- The factors of \( -9 \) are \( -9, -3, -1, 1, 3, 9 \).
- The factors of \( -2 \) are \( -2, -1, 1, 2 \).
- The factors of \( 6 \) are \( 1, 2, 3, 6 \).
- The greatest common factor is \( 1 \).

2. Identify the GCF of the variables \( u^6v \), \( u^2v \), \( uv^2 \), and \( uv \):
- The lowest power of \( u \) is \( u \).
- The lowest power of \( v \) is \( v \).

3. Factor out the GCF:
\[
2u^6v - 9u^2v - 2uv^2 + 6uv = uv(2u^5 - 9u - 2v + 6)
\]

Problem 13: \( -8z^7y + 24z^6x \)

1. Identify the GCF of the coefficients \( -8 \) and \( 24 \):
- The factors of \( -8 \) are \( -8, -4, -2, -1, 1, 2, 4, 8 \).
- The factors of \( 24 \) are \( 1, 2, 3, 4, 6, 8, 12, 24 \).
- The greatest common factor is \( 8 \).

2. Identify the GCF of the variables \( z^7y \) and \( z^6x \):
- The lowest power of \( z \) is \( z^6 \).
- The variables \( y \) and \( x \) are not common.

3. Factor out the GCF:
\[
-8z^7y + 24z^6x = 8z^6(-zy + 3x)
\]

Problem 14: \( 40a^3b^2c^2 - 30a^5c - 20a^3b \)

1. Identify the GCF of the coefficients \( 40 \), \( -30 \), and \( -20 \):
- The factors of \( 40 \) are \( 1, 2, 4, 5, 8, 10, 20, 40 \).
- The factors of \( -30 \) are \( -30, -15, -10, -6, -5, -3, -2, -1, 1, 2, 3, 5, 6, 10, 15, 30 \).
- The factors of \( -20 \) are \( -20, -10, -5, -4, -2, -1, 1, 2, 4, 5, 10, 20 \).
- The greatest common factor is \( 10 \).

2. Identify the GCF of the variables \( a^3b^2c^2 \), \( a^5c \), and \( a^3b \):
- The lowest power of \( a \) is \( a^3 \).
- The lowest power of \( b \) is \( b \).
- The lowest power of \( c \) is \( 1 \) (since \( c \) is not present in the third term).

3. Factor out the GCF:
\[
40a^3b^2c^2 - 30a^5c - 20a^3b = 10a^3(4b^2c^2 - 3a^2c - 2b)
\]

Final Answer:

\[
\boxed{
\begin{aligned}
1) & \quad 8(4b + 1) \\
2) & \quad 5(-n + 3) \\
3) & \quad 5a^2(7a^2 + 6) \\
4) & \quad 6n^2(n + 8) \\
5) & \quad 9x^2(3x^2y + 5) \\
6) & \quad 8ab^3(b + 4) \\
7) & \quad 3(10x^2 + 8x + 7) \\
8) & \quad 7k(3k^3 + 3k - 1) \\
9) & \quad 8(9n^3 + n^2 + 5n - 7) \\
10) & \quad 2m^3(9 - 5m + 6m^2 + 6m^6) \\
11) & \quad 10u(4v^3 + 10u + 7v) \\
12) & \quad uv(2u^5 - 9u - 2v + 6) \\
13) & \quad 8z^6(-zy + 3x) \\
14) & \quad 10a^3(4b^2c^2 - 3a^2c - 2b)
\end{aligned}
}
\]