Grade 9 Factoring Trinomials Worksheets 2024 - Free Printable

Let’s solve each problem step by step.

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Problems 1–20: Factor each trinomial

We’re looking for two binomials that multiply to give the trinomial. For trinomials like \(x^2 + bx + c\), we look for two numbers that:
- Multiply to c
- Add to b

For trinomials with a leading coefficient (like \(6x^2 + x - 12\)), we use “AC method” or trial and error with factors of the first and last terms.

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1) \(x^2 + 8x + 15\)
Need two numbers that multiply to 15, add to 8 → 3 and 5
→ \((x + 3)(x + 5)\)

2) \(x^2 - 5x + 6\)
Multiply to 6, add to -5 → -2 and -3
→ \((x - 2)(x - 3)\)

3) \(x^2 + 6x + 8\)
Multiply to 8, add to 6 → 2 and 4
→ \((x + 2)(x + 4)\)

4) \(x^2 - 6x + 8\)
Multiply to 8, add to -6 → -2 and -4
→ \((x - 2)(x - 4)\)

5) \(x^2 - 8x + 16\)
Multiply to 16, add to -8 → -4 and -4
→ \((x - 4)^2\) or \((x - 4)(x - 4)\)

6) \(x^2 - 7x + 12\)
Multiply to 12, add to -7 → -3 and -4
→ \((x - 3)(x - 4)\)

7) \(x^2 + 11x + 18\)
Multiply to 18, add to 11 → 2 and 9
→ \((x + 2)(x + 9)\)

8) \(x^2 + 2x - 24\)
Multiply to -24, add to 2 → 6 and -4
→ \((x + 6)(x - 4)\)

9) \(x^2 + 4x - 12\)
Multiply to -12, add to 4 → 6 and -2
→ \((x + 6)(x - 2)\)

10) \(x^2 - 10x + 9\)
Multiply to 9, add to -10 → -1 and -9
→ \((x - 1)(x - 9)\)

11) \(x^2 + 5x - 14\)
Multiply to -14, add to 5 → 7 and -2
→ \((x + 7)(x - 2)\)

12) \(x^2 - 6x - 27\)
Multiply to -27, add to -6 → -9 and 3
→ \((x - 9)(x + 3)\)

13) \(x^2 - 11x - 42\)
Multiply to -42, add to -11 → -14 and 3
→ \((x - 14)(x + 3)\)

14) \(x^2 + 22x + 121\)
Multiply to 121, add to 22 → 11 and 11
→ \((x + 11)^2\)

15) \(6x^2 + x - 12\)
Use AC method: A=6, C=-12 → AC = -72
Find two numbers that multiply to -72, add to 1 → 9 and -8
Rewrite middle term: \(6x^2 + 9x - 8x - 12\)
Group: \((6x^2 + 9x) + (-8x - 12)\)
Factor: \(3x(2x + 3) -4(2x + 3)\)
→ \((3x - 4)(2x + 3)\)

16) \(x^2 - 17x + 30\)
Multiply to 30, add to -17 → -15 and -2
→ \((x - 15)(x - 2)\)

17) \(3x^2 + 11x - 4\)
AC = 3 * -4 = -12
Numbers that multiply to -12, add to 11 → 12 and -1
Rewrite: \(3x^2 + 12x - x - 4\)
Group: \(3x(x + 4) -1(x + 4)\)
→ \((3x - 1)(x + 4)\)

18) \(10x^2 + 33x - 7\)
AC = 10 * -7 = -70
Numbers that multiply to -70, add to 33 → 35 and -2
Rewrite: \(10x^2 + 35x - 2x - 7\)
Group: \(5x(2x + 7) -1(2x + 7)\)
→ \((5x - 1)(2x + 7)\)

19) \(x^2 + 24x + 144\)
Multiply to 144, add to 24 → 12 and 12
→ \((x + 12)^2\)

20) \(8x^2 + 10x - 3\)
AC = 8 * -3 = -24
Numbers that multiply to -24, add to 10 → 12 and -2
Rewrite: \(8x^2 + 12x - 2x - 3\)
Group: \(4x(2x + 3) -1(2x + 3)\)
→ \((4x - 1)(2x + 3)\)

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Problems 21–23: Solve each problem

These involve factoring area expressions and dividing by one dimension to find the other.

Recall:
- Area of rectangle = length × width
- Area of parallelogram = base × height

So if you have area and one side, divide area by that side to get the other.

But since these are polynomials, we’ll factor the area expression and cancel out the known side.

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21) Area = \(x^2 + 2x - 24\), Width = \(x - 4\). Find Length.

Factor area:
\(x^2 + 2x - 24\) → need two numbers that multiply to -24, add to 2 → 6 and -4
→ \((x + 6)(x - 4)\)

Width is \(x - 4\), so length must be \(x + 6\)

✔ Check: \((x - 4)(x + 6) = x^2 + 6x - 4x - 24 = x^2 + 2x - 24\) ✔️

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22) Area = \(8x^2 + 2x - 6\), Height = \(2x + 2\). Find Base.

First, factor area:
\(8x^2 + 2x - 6\)

Factor out GCF first? All coefficients divisible by 2 →
= \(2(4x^2 + x - 3)\)

Now factor \(4x^2 + x - 3\):
AC = 4 * -3 = -12
Numbers that multiply to -12, add to 1 → 4 and -3
Rewrite: \(4x^2 + 4x - 3x - 3\)
Group: \(4x(x + 1) -3(x + 1)\)
→ \((4x - 3)(x + 1)\)

So full factorization: \(2(4x - 3)(x + 1)\)

Height is \(2x + 2 = 2(x + 1)\)

So base = Area ÷ Height = \(\frac{2(4x - 3)(x + 1)}{2(x + 1)} = 4x - 3\)

✔ Check: \((2x + 2)(4x - 3) = 8x^2 -6x + 8x -6 = 8x^2 + 2x -6\) ✔️

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23) Area = \(18x^2 + 9x - 2\), Width = \(6x - 1\). Find Length.

Factor area: \(18x^2 + 9x - 2\)

AC = 18 * -2 = -36
Numbers that multiply to -36, add to 9 → 12 and -3
Rewrite: \(18x^2 + 12x - 3x - 2\)
Group: \(6x(3x + 2) -1(3x + 2)\)
→ \((6x - 1)(3x + 2)\)

Width is \(6x - 1\), so length is \(3x + 2\)

✔ Check: \((6x - 1)(3x + 2) = 18x^2 + 12x - 3x - 2 = 18x^2 + 9x - 2\) ✔️

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Final Answer:

1) (x + 3)(x + 5)
2) (x - 2)(x - 3)
3) (x + 2)(x + 4)
4) (x - 2)(x - 4)
5) (x - 4)(x - 4) or (x - 4)²
6) (x - 3)(x - 4)
7) (x + 2)(x + 9)
8) (x + 6)(x - 4)
9) (x + 6)(x - 2)
10) (x - 1)(x - 9)
11) (x + 7)(x - 2)
12) (x - 9)(x + 3)
13) (x - 14)(x + 3)
14) (x + 11)(x + 11) or (x + 11)²
15) (3x - 4)(2x + 3)
16) (x - 15)(x - 2)
17) (3x - 1)(x + 4)
18) (5x - 1)(2x + 7)
19) (x + 12)(x + 12) or (x + 12)²
20) (4x - 1)(2x + 3)
21) x + 6
22) 4x - 3
23) 3x + 2