Factoring Worksheets - Math Monks - Free Printable

We are given 10 quadratic (or higher-degree) expressions to factor completely, where the leading coefficient a ≠ 1. We’ll use factoring techniques like:

- Factoring out the GCF (Greatest Common Factor) first if possible.
- AC method (also called “splitting the middle term”) for quadratics: multiply a·c, find two numbers that multiply to a·c and add to b, then split the middle term and factor by grouping.
- For trinomials with two variables, treat as quadratic in one variable.

Let’s solve each one step by step.

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① 3x² - 16x + 20

- a = 3, b = -16, c = 20
- a·c = 3×20 = 60
- Find two numbers that multiply to 60 and add to -16 → -6 and -10

Split middle term:
> 3x² - 6x - 10x + 20
Group:
> (3x² - 6x) + (-10x + 20)
> = 3x(x - 2) - 10(x - 2)
> = (3x - 10)(x - 2)

✔ Answer: (3x - 10)(x - 2)

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② 10x² - 27x + 5

- a = 10, b = -27, c = 5
- a·c = 50
- Need two numbers multiplying to 50 and adding to -27 → -2 and -25

Split:
> 10x² - 25x - 2x + 5
Group:
> (10x² - 25x) + (-2x + 5)
> = 5x(2x - 5) -1(2x - 5)
> = (5x - 1)(2x - 5)

✔ Answer: (5x - 1)(2x - 5)

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③ 18x³ + 33x² - 30x

First, factor out GCF.

All terms divisible by 3x:
> 3x(6x² + 11x - 10)

Now factor 6x² + 11x - 10:

- a = 6, b = 11, c = -10
- a·c = -60
- Need two numbers multiplying to -60 and adding to 11 → 15 and -4

Split:
> 6x² + 15x - 4x - 10
Group:
> (6x² + 15x) + (-4x - 10)
> = 3x(2x + 5) -2(2x + 5)
> = (3x - 2)(2x + 5)

So overall:
> 3x(3x - 2)(2x + 5)

✔ Answer: 3x(3x - 2)(2x + 5)

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④ 24x² - 52xy + 8y²

First, factor out GCF.

All terms divisible by 4:
> 4(6x² - 13xy + 2y²)

Now factor 6x² - 13xy + 2y²

Treat as quadratic in x, coefficients involve y.

- a = 6, b = -13y, c = 2y²
- a·c = 12y²
- Need two terms multiplying to 12y² and adding to -13y → -12y and -y

Split:
> 6x² - 12xy - xy + 2y²
Group:
> (6x² - 12xy) + (-xy + 2y²)
> = 6x(x - 2y) - y(x - 2y)
> = (6x - y)(x - 2y)

So overall:
> 4(6x - y)(x - 2y)

✔ Answer: 4(6x - y)(x - 2y)

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⑤ 5a² - 11a + 6

- a = 5, b = -11, c = 6
- a·c = 30
- Numbers multiplying to 30, adding to -11 → -5 and -6

Split:
> 5a² - 5a - 6a + 6
Group:
> (5a² - 5a) + (-6a + 6)
> = 5a(a - 1) -6(a - 1)
> = (5a - 6)(a - 1)

✔ Answer: (5a - 6)(a - 1)

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⑥ 8p² - 20p - 72

First, factor out GCF.

All divisible by 4:
> 4(2p² - 5p - 18)

Factor 2p² - 5p - 18:

- a = 2, b = -5, c = -18
- a·c = -36
- Need two numbers multiplying to -36, adding to -5 → -9 and 4

Split:
> 2p² - 9p + 4p - 18
Group:
> (2p² - 9p) + (4p - 18)
> = p(2p - 9) + 2(2p - 9)
> = (p + 2)(2p - 9)

So overall:
> 4(p + 2)(2p - 9)

✔ Answer: 4(p + 2)(2p - 9)

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⑦ 6n² + 45n + 84

First, factor out GCF.

All divisible by 3:
> 3(2n² + 15n + 28)

Factor 2n² + 15n + 28:

- a = 2, b = 15, c = 28
- a·c = 56
- Need two numbers multiplying to 56, adding to 15 → 7 and 8

Split:
> 2n² + 7n + 8n + 28
Group:
> (2n² + 7n) + (8n + 28)
> = n(2n + 7) + 4(2n + 7)
> = (n + 4)(2n + 7)

So overall:
> 3(n + 4)(2n + 7)

✔ Answer: 3(n + 4)(2n + 7)

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⑧ 10m² + 23m + 6

- a = 10, b = 23, c = 6
- a·c = 60
- Need two numbers multiplying to 60, adding to 23 → 20 and 3

Split:
> 10m² + 20m + 3m + 6
Group:
> (10m² + 20m) + (3m + 6)
> = 10m(m + 2) + 3(m + 2)
> = (10m + 3)(m + 2)

✔ Answer: (10m + 3)(m + 2)

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⑨ 16a² + 60a - 100

First, factor out GCF.

All divisible by 4:
> 4(4a² + 15a - 25)

Factor 4a² + 15a - 25:

- a = 4, b = 15, c = -25
- a·c = -100
- Need two numbers multiplying to -100, adding to 15 → 20 and -5

Split:
> 4a² + 20a - 5a - 25
Group:
> (4a² + 20a) + (-5a - 25)
> = 4a(a + 5) -5(a + 5)
> = (4a - 5)(a + 5)

So overall:
> 4(4a - 5)(a + 5)

✔ Answer: 4(4a - 5)(a + 5)

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⑩ 5u² + 31uv - 28v²

Treat as quadratic in u, coefficients involve v.

- a = 5, b = 31v, c = -28v²
- a·c = -140v²
- Need two terms multiplying to -140v², adding to 31v → 35v and -4v

Split:
> 5u² + 35uv - 4uv - 28v²
Group:
> (5u² + 35uv) + (-4uv - 28v²)
> = 5u(u + 7v) -4v(u + 7v)
> = (5u - 4v)(u + 7v)

✔ Answer: (5u - 4v)(u + 7v)

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## ✔ Final Answers:

① (3x - 10)(x - 2)
② (5x - 1)(2x - 5)
③ 3x(3x - 2)(2x + 5)
④ 4(6x - y)(x - 2y)
⑤ (5a - 6)(a - 1)
⑥ 4(p + 2)(2p - 9)
⑦ 3(n + 4)(2n + 7)
⑧ (10m + 3)(m + 2)
⑨ 4(4a - 5)(a + 5)
⑩ (5u - 4v)(u + 7v)

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