Factoring Trinomials worksheet with 18 algebra problems for practice.

To solve the problem of factoring each trinomial, we will go through each expression step by step. Factoring trinomials involves expressing them as a product of two binomials (or other simpler factors). Let's solve each one:

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1) \( x^2 + 7x + 12 \)

We need to find two numbers that multiply to \( 12 \) (the constant term) and add up to \( 7 \) (the coefficient of \( x \)).

- The pairs of factors of \( 12 \) are: \( (1, 12), (2, 6), (3, 4) \).
- The pair \( (3, 4) \) adds up to \( 7 \).

Thus, we can write:
\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\]

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2) \( 9x^2 - 34x + 21 \)

This is a quadratic trinomial with a leading coefficient other than 1. We use the "ac-method":

1. Multiply the leading coefficient (\( 9 \)) by the constant term (\( 21 \)):
\[
9 \cdot 21 = 189
\]

2. Find two numbers that multiply to \( 189 \) and add up to \( -34 \):
- The pairs of factors of \( 189 \) are: \( (-1, -189), (-3, -63), (-7, -27), (-9, -21) \).
- The pair \( (-9, -21) \) adds up to \( -34 \).

3. Rewrite the middle term using these numbers:
\[
9x^2 - 34x + 21 = 9x^2 - 9x - 21x + 21
\]

4. Factor by grouping:
\[
(9x^2 - 9x) + (-21x + 21) = 9x(x - 1) - 21(x - 1)
\]

5. Factor out the common binomial factor \( (x - 1) \):
\[
9x(x - 1) - 21(x - 1) = (9x - 21)(x - 1)
\]

6. Simplify the first binomial:
\[
9x - 21 = 3(3x - 7)
\]

Thus, the factored form is:
\[
9x^2 - 34x + 21 = (3x - 7)(3x - 3)
\]

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3) \( x^2 - x - 90 \)

We need to find two numbers that multiply to \( -90 \) and add up to \( -1 \).

- The pairs of factors of \( -90 \) are: \( (1, -90), (-1, 90), (2, -45), (-2, 45), (3, -30), (-3, 30), (5, -18), (-5, 18), (6, -15), (-6, 15), (9, -10), (-9, 10) \).
- The pair \( (9, -10) \) adds up to \( -1 \).

Thus, we can write:
\[
x^2 - x - 90 = (x + 9)(x - 10)
\]

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4) \( x^2 - 8x + 12 \)

We need to find two numbers that multiply to \( 12 \) and add up to \( -8 \).

- The pairs of factors of \( 12 \) are: \( (1, 12), (-1, -12), (2, 6), (-2, -6), (3, 4), (-3, -4) \).
- The pair \( (-2, -6) \) adds up to \( -8 \).

Thus, we can write:
\[
x^2 - 8x + 12 = (x - 2)(x - 6)
\]

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5) \( x^2 - 3x + 2 \)

We need to find two numbers that multiply to \( 2 \) and add up to \( -3 \).

- The pairs of factors of \( 2 \) are: \( (1, 2), (-1, -2) \).
- The pair \( (-1, -2) \) adds up to \( -3 \).

Thus, we can write:
\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]

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6) \( x^2 - 10x + 24 \)

We need to find two numbers that multiply to \( 24 \) and add up to \( -10 \).

- The pairs of factors of \( 24 \) are: \( (1, 24), (-1, -24), (2, 12), (-2, -12), (3, 8), (-3, -8), (4, 6), (-4, -6) \).
- The pair \( (-4, -6) \) adds up to \( -10 \).

Thus, we can write:
\[
x^2 - 10x + 24 = (x - 4)(x - 6)
\]

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7) \( 4x^2 - 4x - 120 \)

First, factor out the greatest common factor (GCF), which is \( 4 \):
\[
4x^2 - 4x - 120 = 4(x^2 - x - 30)
\]

Now, factor \( x^2 - x - 30 \):

- We need two numbers that multiply to \( -30 \) and add up to \( -1 \).
- The pairs of factors of \( -30 \) are: \( (1, -30), (-1, 30), (2, -15), (-2, 15), (3, -10), (-3, 10), (5, -6), (-5, 6) \).
- The pair \( (-6, 5) \) adds up to \( -1 \).

Thus:
\[
x^2 - x - 30 = (x - 6)(x + 5)
\]

So:
\[
4x^2 - 4x - 120 = 4(x - 6)(x + 5)
\]

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8) \( x^2 + 21x + 20 \)

We need to find two numbers that multiply to \( 20 \) and add up to \( 21 \).

- The pairs of factors of \( 20 \) are: \( (1, 20), (2, 10), (4, 5) \).
- The pair \( (1, 20) \) adds up to \( 21 \).

Thus, we can write:
\[
x^2 + 21x + 20 = (x + 1)(x + 20)
\]

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9) \( x^2 - 12x + 20 \)

We need to find two numbers that multiply to \( 20 \) and add up to \( -12 \).

- The pairs of factors of \( 20 \) are: \( (1, 20), (-1, -20), (2, 10), (-2, -10), (4, 5), (-4, -5) \).
- The pair \( (-2, -10) \) adds up to \( -12 \).

Thus, we can write:
\[
x^2 - 12x + 20 = (x - 2)(x - 10)
\]

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10) \( 4x^2 - 20x - 24 \)

First, factor out the GCF, which is \( 4 \):
\[
4x^2 - 20x - 24 = 4(x^2 - 5x - 6)
\]

Now, factor \( x^2 - 5x - 6 \):

- We need two numbers that multiply to \( -6 \) and add up to \( -5 \).
- The pairs of factors of \( -6 \) are: \( (1, -6), (-1, 6), (2, -3), (-2, 3) \).
- The pair \( (-6, 1) \) adds up to \( -5 \).

Thus:
\[
x^2 - 5x - 6 = (x - 6)(x + 1)
\]

So:
\[
4x^2 - 20x - 24 = 4(x - 6)(x + 1)
\]

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11) \( x^2 - 12x + 35 \)

We need to find two numbers that multiply to \( 35 \) and add up to \( -12 \).

- The pairs of factors of \( 35 \) are: \( (1, 35), (-1, -35), (5, 7), (-5, -7) \).
- The pair \( (-5, -7) \) adds up to \( -12 \).

Thus, we can write:
\[
x^2 - 12x + 35 = (x - 5)(x - 7)
\]

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12) \( x^2 + 12x - 13 \)

We need to find two numbers that multiply to \( -13 \) and add up to \( 12 \).

- The pairs of factors of \( -13 \) are: \( (1, -13), (-1, 13) \).
- The pair \( (13, -1) \) adds up to \( 12 \).

Thus, we can write:
\[
x^2 + 12x - 13 = (x + 13)(x - 1)
\]

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13) \( x^2 - 10x - 39 \)

We need to find two numbers that multiply to \( -39 \) and add up to \( -10 \).

- The pairs of factors of \( -39 \) are: \( (1, -39), (-1, 39), (3, -13), (-3, 13) \).
- The pair \( (-13, 3) \) adds up to \( -10 \).

Thus, we can write:
\[
x^2 - 10x - 39 = (x - 13)(x + 3)
\]

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14) \( x^2 + 15x - 34 \)

We need to find two numbers that multiply to \( -34 \) and add up to \( 15 \).

- The pairs of factors of \( -34 \) are: \( (1, -34), (-1, 34), (2, -17), (-2, 17) \).
- The pair \( (17, -2) \) adds up to \( 15 \).

Thus, we can write:
\[
x^2 + 15x - 34 = (x + 17)(x - 2)
\]

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15) \( x^2 - 4 \)

This is a difference of squares:
\[
x^2 - 4 = (x - 2)(x + 2)
\]

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16) \( x^2 - 64 \)

This is also a difference of squares:
\[
x^2 - 64 = (x - 8)(x + 8)
\]

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17) \( 2x^3 + 8x^2 - 24x \)

First, factor out the GCF, which is \( 2x \):
\[
2x^3 + 8x^2 - 24x = 2x(x^2 + 4x - 12)
\]

Now, factor \( x^2 + 4x - 12 \):

- We need two numbers that multiply to \( -12 \) and add up to \( 4 \).
- The pairs of factors of \( -12 \) are: \( (1, -12), (-1, 12), (2, -6), (-2, 6), (3, -4), (-3, 4) \).
- The pair \( (6, -2) \) adds up to \( 4 \).

Thus:
\[
x^2 + 4x - 12 = (x + 6)(x - 2)
\]

So:
\[
2x^3 + 8x^2 - 24x = 2x(x + 6)(x - 2)
\]

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18) \( x^4 + x^3 - 2x^2 \)

First, factor out the GCF, which is \( x^2 \):
\[
x^4 + x^3 - 2x^2 = x^2(x^2 + x - 2)
\]

Now, factor \( x^2 + x - 2 \):

- We need two numbers that multiply to \( -2 \) and add up to \( 1 \).
- The pairs of factors of \( -2 \) are: \( (1, -2), (-1, 2) \).
- The pair \( (2, -1) \) adds up to \( 1 \).

Thus:
\[
x^2 + x - 2 = (x + 2)(x - 1)
\]

So:
\[
x^4 + x^3 - 2x^2 = x^2(x + 2)(x - 1)
\]

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Final Answers:

\[
\boxed{
\begin{aligned}
1) & \ (x + 3)(x + 4) \\
2) & \ (3x - 7)(3x - 3) \\
3) & \ (x + 9)(x - 10) \\
4) & \ (x - 2)(x - 6) \\
5) & \ (x - 1)(x - 2) \\
6) & \ (x - 4)(x - 6) \\
7) & \ 4(x - 6)(x + 5) \\
8) & \ (x + 1)(x + 20) \\
9) & \ (x - 2)(x - 10) \\
10) & \ 4(x - 6)(x + 1) \\
11) & \ (x - 5)(x - 7) \\
12) & \ (x + 13)(x - 1) \\
13) & \ (x - 13)(x + 3) \\
14) & \ (x + 17)(x - 2) \\
15) & \ (x - 2)(x + 2) \\
16) & \ (x - 8)(x + 8) \\
17) & \ 2x(x + 6)(x - 2) \\
18) & \ x^2(x + 2)(x - 1)
\end{aligned}
}
\]