Factoring Polynomials Worksheets with Answer Key - Free Printable

To solve the problem of factoring each polynomial expression completely, we will follow these steps:

1. Identify the Greatest Common Factor (GCF): Factor out the GCF from all terms in the polynomial.
2. Factor quadratic expressions: If the polynomial is a quadratic expression, factor it into two binomials.
3. Check for further factorization: Ensure that the resulting factors cannot be factored further.

Let's go through each expression step by step.

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Problem 1: \( 2ab^2 - 14ab \)

- Step 1: Identify the GCF.
- The terms are \( 2ab^2 \) and \( -14ab \).
- The GCF of the coefficients \( 2 \) and \( -14 \) is \( 2 \).
- The GCF of the variables \( ab^2 \) and \( ab \) is \( ab \).
- Therefore, the GCF is \( 2ab \).

- Step 2: Factor out the GCF.
\[
2ab^2 - 14ab = 2ab(b - 7)
\]

- Final Answer:
\[
\boxed{2ab(b - 7)}
\]

---

Problem 2: \( 5a^2 + 55a \)

- Step 1: Identify the GCF.
- The terms are \( 5a^2 \) and \( 55a \).
- The GCF of the coefficients \( 5 \) and \( 55 \) is \( 5 \).
- The GCF of the variables \( a^2 \) and \( a \) is \( a \).
- Therefore, the GCF is \( 5a \).

- Step 2: Factor out the GCF.
\[
5a^2 + 55a = 5a(a + 11)
\]

- Final Answer:
\[
\boxed{5a(a + 11)}
\]

---

Problem 3: \( 18v^2 + 114v - 84 \)

- Step 1: Identify the GCF.
- The terms are \( 18v^2 \), \( 114v \), and \( -84 \).
- The GCF of the coefficients \( 18 \), \( 114 \), and \( -84 \) is \( 6 \).

- Step 2: Factor out the GCF.
\[
18v^2 + 114v - 84 = 6(3v^2 + 19v - 14)
\]

- Step 3: Factor the quadratic expression \( 3v^2 + 19v - 14 \).
- We need two numbers that multiply to \( 3 \times (-14) = -42 \) and add to \( 19 \).
- These numbers are \( 21 \) and \( -2 \).
- Rewrite the middle term using these numbers:
\[
3v^2 + 19v - 14 = 3v^2 + 21v - 2v - 14
\]
- Factor by grouping:
\[
3v^2 + 21v - 2v - 14 = 3v(v + 7) - 2(v + 7)
\]
\[
= (3v - 2)(v + 7)
\]

- Final Answer:
\[
\boxed{6(3v - 2)(v + 7)}
\]

---

Problem 4: \( 8x^2 + 16x \)

- Step 1: Identify the GCF.
- The terms are \( 8x^2 \) and \( 16x \).
- The GCF of the coefficients \( 8 \) and \( 16 \) is \( 8 \).
- The GCF of the variables \( x^2 \) and \( x \) is \( x \).
- Therefore, the GCF is \( 8x \).

- Step 2: Factor out the GCF.
\[
8x^2 + 16x = 8x(x + 2)
\]

- Final Answer:
\[
\boxed{8x(x + 2)}
\]

---

Problem 5: \( 25mp^2 - 45mp \)

- Step 1: Identify the GCF.
- The terms are \( 25mp^2 \) and \( -45mp \).
- The GCF of the coefficients \( 25 \) and \( -45 \) is \( 5 \).
- The GCF of the variables \( mp^2 \) and \( mp \) is \( mp \).
- Therefore, the GCF is \( 5mp \).

- Step 2: Factor out the GCF.
\[
25mp^2 - 45mp = 5mp(5p - 9)
\]

- Final Answer:
\[
\boxed{5mp(5p - 9)}
\]

---

Problem 6: \( 16c^2 + 48c \)

- Step 1: Identify the GCF.
- The terms are \( 16c^2 \) and \( 48c \).
- The GCF of the coefficients \( 16 \) and \( 48 \) is \( 16 \).
- The GCF of the variables \( c^2 \) and \( c \) is \( c \).
- Therefore, the GCF is \( 16c \).

- Step 2: Factor out the GCF.
\[
16c^2 + 48c = 16c(c + 3)
\]

- Final Answer:
\[
\boxed{16c(c + 3)}
\]

---

Problem 7: \( m^2y + 6myn + 8n^2y \)

- Step 1: Identify the GCF.
- The terms are \( m^2y \), \( 6myn \), and \( 8n^2y \).
- The GCF of the coefficients \( 1 \), \( 6 \), and \( 8 \) is \( 1 \).
- The GCF of the variables \( m^2y \), \( mny \), and \( n^2y \) is \( y \).
- Therefore, the GCF is \( y \).

- Step 2: Factor out the GCF.
\[
m^2y + 6myn + 8n^2y = y(m^2 + 6mn + 8n^2)
\]

- Step 3: Factor the quadratic expression \( m^2 + 6mn + 8n^2 \).
- We need two numbers that multiply to \( 1 \times 8 = 8 \) and add to \( 6 \).
- These numbers are \( 4 \) and \( 2 \).
- Rewrite the quadratic expression:
\[
m^2 + 6mn + 8n^2 = (m + 4n)(m + 2n)
\]

- Final Answer:
\[
\boxed{y(m + 4n)(m + 2n)}
\]

---

Problem 8: \( -2x^2 + 8xy + 64y^2 \)

- Step 1: Identify the GCF.
- The terms are \( -2x^2 \), \( 8xy \), and \( 64y^2 \).
- The GCF of the coefficients \( -2 \), \( 8 \), and \( 64 \) is \( -2 \).
- There are no common variables among all terms.
- Therefore, the GCF is \( -2 \).

- Step 2: Factor out the GCF.
\[
-2x^2 + 8xy + 64y^2 = -2(x^2 - 4xy - 32y^2)
\]

- Step 3: Factor the quadratic expression \( x^2 - 4xy - 32y^2 \).
- We need two numbers that multiply to \( 1 \times (-32) = -32 \) and add to \( -4 \).
- These numbers are \( -8 \) and \( 4 \).
- Rewrite the quadratic expression:
\[
x^2 - 4xy - 32y^2 = (x - 8y)(x + 4y)
\]

- Final Answer:
\[
\boxed{-2(x - 8y)(x + 4y)}
\]

---

Problem 9: \( -42x^2 + 306xy - 84y^2 \)

- Step 1: Identify the GCF.
- The terms are \( -42x^2 \), \( 306xy \), and \( -84y^2 \).
- The GCF of the coefficients \( -42 \), \( 306 \), and \( -84 \) is \( -6 \).
- There are no common variables among all terms.
- Therefore, the GCF is \( -6 \).

- Step 2: Factor out the GCF.
\[
-42x^2 + 306xy - 84y^2 = -6(7x^2 - 51xy + 14y^2)
\]

- Step 3: Factor the quadratic expression \( 7x^2 - 51xy + 14y^2 \).
- We need two numbers that multiply to \( 7 \times 14 = 98 \) and add to \( -51 \).
- These numbers are \( -49 \) and \( -2 \).
- Rewrite the quadratic expression:
\[
7x^2 - 51xy + 14y^2 = 7x^2 - 49xy - 2xy + 14y^2
\]
- Factor by grouping:
\[
7x^2 - 49xy - 2xy + 14y^2 = 7x(x - 7y) - 2y(x - 7y)
\]
\[
= (7x - 2y)(x - 7y)
\]

- Final Answer:
\[
\boxed{-6(7x - 2y)(x - 7y)}
\]

---

Problem 10: \( -m^2v - 8mvn - 12n^2v \)

- Step 1: Identify the GCF.
- The terms are \( -m^2v \), \( -8mvn \), and \( -12n^2v \).
- The GCF of the coefficients \( -1 \), \( -8 \), and \( -12 \) is \( -1 \).
- The GCF of the variables \( m^2v \), \( mvn \), and \( n^2v \) is \( v \).
- Therefore, the GCF is \( -v \).

- Step 2: Factor out the GCF.
\[
-m^2v - 8mvn - 12n^2v = -v(m^2 + 8mn + 12n^2)
\]

- Step 3: Factor the quadratic expression \( m^2 + 8mn + 12n^2 \).
- We need two numbers that multiply to \( 1 \times 12 = 12 \) and add to \( 8 \).
- These numbers are \( 6 \) and \( 2 \).
- Rewrite the quadratic expression:
\[
m^2 + 8mn + 12n^2 = (m + 6n)(m + 2n)
\]

- Final Answer:
\[
\boxed{-v(m + 6n)(m + 2n)}
\]

---

Problem 11: \( 2y^2 + 28y + 96 \)

- Step 1: Identify the GCF.
- The terms are \( 2y^2 \), \( 28y \), and \( 96 \).
- The GCF of the coefficients \( 2 \), \( 28 \), and \( 96 \) is \( 2 \).

- Step 2: Factor out the GCF.
\[
2y^2 + 28y + 96 = 2(y^2 + 14y + 48)
\]

- Step 3: Factor the quadratic expression \( y^2 + 14y + 48 \).
- We need two numbers that multiply to \( 48 \) and add to \( 14 \).
- These numbers are \( 6 \) and \( 8 \).
- Rewrite the quadratic expression:
\[
y^2 + 14y + 48 = (y + 6)(y + 8)
\]

- Final Answer:
\[
\boxed{2(y + 6)(y + 8)}
\]

---

Problem 12: \( 5b^2k^2 + 25bk^2 - 250k^2 \)

- Step 1: Identify the GCF.
- The terms are \( 5b^2k^2 \), \( 25bk^2 \), and \( -250k^2 \).
- The GCF of the coefficients \( 5 \), \( 25 \), and \( -250 \) is \( 5 \).
- The GCF of the variables \( b^2k^2 \), \( bk^2 \), and \( k^2 \) is \( k^2 \).
- Therefore, the GCF is \( 5k^2 \).

- Step 2: Factor out the GCF.
\[
5b^2k^2 + 25bk^2 - 250k^2 = 5k^2(b^2 + 5b - 50)
\]

- Step 3: Factor the quadratic expression \( b^2 + 5b - 50 \).
- We need two numbers that multiply to \( -50 \) and add to \( 5 \).
- These numbers are \( 10 \) and \( -5 \).
- Rewrite the quadratic expression:
\[
b^2 + 5b - 50 = (b + 10)(b - 5)
\]

- Final Answer:
\[
\boxed{5k^2(b + 10)(b - 5)}
\]

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Final Answers

\[
\boxed{
\begin{aligned}
1. & \quad 2ab(b - 7) \\
2. & \quad 5a(a + 11) \\
3. & \quad 6(3v - 2)(v + 7) \\
4. & \quad 8x(x + 2) \\
5. & \quad 5mp(5p - 9) \\
6. & \quad 16c(c + 3) \\
7. & \quad y(m + 4n)(m + 2n) \\
8. & \quad -2(x - 8y)(x + 4y) \\
9. & \quad -6(7x - 2y)(x - 7y) \\
10. & \quad -v(m + 6n)(m + 2n) \\
11. & \quad 2(y + 6)(y + 8) \\
12. & \quad 5k^2(b + 10)(b - 5)
\end{aligned}
}
\]