Factoring Worksheets - Math Monks - Free Printable

Let's solve each of the factoring problems step by step.

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① $ x^2 + 6x + 8 $

We need two numbers that multiply to 8 and add to 6.

- Factors of 8: (1,8), (2,4)
- 2 and 4 → 2×4 = 8, 2+4 = 6 ✔

So:
$$
x^2 + 6x + 8 = (x + 2)(x + 4)
$$

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② $ x^2 + 7x - 18 $

Need two numbers that multiply to -18 and add to 7.

- Pairs: (1,-18), (-1,18), (2,-9), (-2,9), (3,-6), (-3,6)
- Try: 9 and -2 → 9×(-2) = -18, 9 + (-2) = 7 ✔

So:
$$
x^2 + 7x - 18 = (x + 9)(x - 2)
$$

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③ $ 6x^2 - x - 12 $

Use AC method:
$ a = 6, b = -1, c = -12 $
$ ac = 6 × (-12) = -72 $

Find two numbers that multiply to -72 and add to -1.

Try: -9 and 8 → -9×8 = -72, -9 + 8 = -1 ✔

Now split the middle term:
$$
6x^2 - 9x + 8x - 12
$$

Group:
$$
(6x^2 - 9x) + (8x - 12) = 3x(2x - 3) + 4(2x - 3)
$$

Factor:
$$
(3x + 4)(2x - 3)
$$

✔ Answer: $ (3x + 4)(2x - 3) $

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④ $ x^3 - 3x^2 + 5x - 15 $

Group terms:
$$
(x^3 - 3x^2) + (5x - 15)
$$

Factor each group:
$$
x^2(x - 3) + 5(x - 3)
$$

Common factor: $ (x - 3) $
$$
(x - 3)(x^2 + 5)
$$

✔ Answer: $ (x - 3)(x^2 + 5) $

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⑤ $ 50x^2 - 72 $

This is a difference of squares after factoring out GCF.

First, factor out GCF: 2
$$
2(25x^2 - 36)
$$

Now: $ 25x^2 - 36 = (5x)^2 - (6)^2 = (5x - 6)(5x + 6) $

So:
$$
50x^2 - 72 = 2(5x - 6)(5x + 6)
$$

✔ Answer: $ 2(5x - 6)(5x + 6) $

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⑥ $ m^2x + 6mnx + 8n^2x $

All terms have x, so factor out x:
$$
x(m^2 + 6mn + 8n^2)
$$

Now factor the quadratic in parentheses:
$ m^2 + 6mn + 8n^2 $

Look for two numbers that multiply to 8 and add to 6 → 2 and 4

$$
= x(m + 2n)(m + 4n)
$$

✔ Answer: $ x(m + 2n)(m + 4n) $

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⑦ $ -m^2v - 8mvn - 12n^2v $

All terms have -v, so factor out -v:
$$
-v(m^2 + 8mn + 12n^2)
$$

Now factor $ m^2 + 8mn + 12n^2 $

Find two numbers that multiply to 12 and add to 8 → 6 and 2

$$
= -v(m + 6n)(m + 2n)
$$

✔ Answer: $ -v(m + 6n)(m + 2n) $

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⑧ $ 7x^2y - 27xy^2 + 18y^3 $

All terms have y, so factor out y:
$$
y(7x^2 - 27xy + 18y^2)
$$

Now factor $ 7x^2 - 27xy + 18y^2 $

Use AC method: $ a = 7, b = -27, c = 18 $

$ ac = 7×18 = 126 $

Find two numbers that multiply to 126 and add to -27

Try: -21 and -6 → -21×-6 = 126, -21 + (-6) = -27 ✔

Split middle term:
$$
7x^2 - 21xy - 6xy + 18y^2
$$

Group:
$$
(7x^2 - 21xy) + (-6xy + 18y^2) = 7x(x - 3y) -6y(x - 3y)
$$

Factor:
$$
(7x - 6y)(x - 3y)
$$

So full expression:
$$
y(7x - 6y)(x - 3y)
$$

✔ Answer: $ y(7x - 6y)(x - 3y) $

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⑨ $ 4x^2 + 4xy - 35y^2 $

Quadratic in $ x $: $ 4x^2 + 4xy - 35y^2 $

Use AC method: $ a = 4, b = 4, c = -35 $

$ ac = 4 × (-35) = -140 $

Find two numbers that multiply to -140 and add to 4

Try: 14 and -10 → 14×(-10) = -140, 14 + (-10) = 4 ✔

Split:
$$
4x^2 + 14xy - 10xy - 35y^2
$$

Group:
$$
(4x^2 + 14xy) + (-10xy - 35y^2) = 2x(2x + 7y) -5y(2x + 7y)
$$

Factor:
$$
(2x - 5y)(2x + 7y)
$$

✔ Answer: $ (2x - 5y)(2x + 7y) $

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⑩ $ -42x^2 + 306xy - 84y^2 $

First, factor out GCF. All coefficients divisible by -6:

$$
-6(7x^2 - 51xy + 14y^2)
$$

Now factor $ 7x^2 - 51xy + 14y^2 $

Use AC method: $ a = 7, b = -51, c = 14 $

$ ac = 7×14 = 98 $

Need two numbers that multiply to 98 and add to -51

Try: -49 and -2 → -49×-2 = 98, -49 + (-2) = -51 ✔

Split:
$$
7x^2 - 49xy - 2xy + 14y^2
$$

Group:
$$
(7x^2 - 49xy) + (-2xy + 14y^2) = 7x(x - 7y) -2y(x - 7y)
$$

Factor:
$$
(7x - 2y)(x - 7y)
$$

Now include the GCF:
$$
-6(7x - 2y)(x - 7y)
$$

✔ Answer: $ -6(7x - 2y)(x - 7y) $

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✔ Final Answers:

1. $ (x + 2)(x + 4) $
2. $ (x + 9)(x - 2) $
3. $ (3x + 4)(2x - 3) $
4. $ (x - 3)(x^2 + 5) $
5. $ 2(5x - 6)(5x + 6) $
6. $ x(m + 2n)(m + 4n) $
7. $ -v(m + 6n)(m + 2n) $
8. $ y(7x - 6y)(x - 3y) $
9. $ (2x - 5y)(2x + 7y) $
10. $ -6(7x - 2y)(x - 7y) $

Let me know if you'd like these explained in more detail!