Factoring Worksheets - Math Monks - Free Printable

Let’s solve each problem step by step. We’re factoring algebraic expressions — that means rewriting them as a product of simpler expressions (like breaking apart numbers into factors, but with variables).

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Problem ①: x² + 6x + 8

We need two numbers that:
- Multiply to 8 (the constant term)
- Add to 6 (the coefficient of x)

Factors of 8:
1 × 8 → 1+8 = 9 ✘
2 × 4 → 2+4 = 6 ✔

So:
→ (x + 2)(x + 4)

✔ Check: (x+2)(x+4) = x² + 4x + 2x + 8 = x² + 6x + 8 ✔️

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Problem ②: x² + 7x - 18

Need two numbers that:
- Multiply to -18
- Add to 7

Try pairs:
9 and -2 → 9 × (-2) = -18, 9 + (-2) = 7 ✔

So:
→ (x + 9)(x - 2)

✔ Check: (x+9)(x-2) = x² -2x +9x -18 = x² +7x -18 ✔️

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Problem ③: 6x² - x - 12

This is trickier because the x² has a coefficient (6). Use “AC method”:

Multiply A×C = 6 × (-12) = -72
Find two numbers that multiply to -72 and add to -1 (middle term coefficient).

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

Now split middle term:
6x² + 8x - 9x - 12

Group:
(6x² + 8x) + (-9x - 12)
= 2x(3x + 4) -3(3x + 4)
→ (2x - 3)(3x + 4)

✔ Check: (2x-3)(3x+4) = 6x² +8x -9x -12 = 6x² -x -12 ✔️

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Problem ④: x³ - 3x² + 5x - 15

Group terms:
(x³ - 3x²) + (5x - 15)
= x²(x - 3) + 5(x - 3)
→ (x² + 5)(x - 3)

✔ Check: (x²+5)(x-3) = x³ -3x² +5x -15 ✔️

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Problem ⑤: 50x² - 72

First, factor out GCF: both divisible by 2
→ 2(25x² - 36)

Now, 25x² - 36 is difference of squares:
(5x)² - (6)² → (5x - 6)(5x + 6)

Final:
→ 2(5x - 6)(5x + 6)

✔ Check: 2*(25x² - 36) = 50x² - 72 ✔️

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Problem ⑥: m²x + 6mnx + 8n²x

All terms have ‘x’ → factor it out first:
→ x(m² + 6mn + 8n²)

Now factor inside: m² + 6mn + 8n²
Think: what two numbers multiply to 8n² and add to 6n?
→ 2n and 4n → (m + 2n)(m + 4n)

Final:
→ x(m + 2n)(m + 4n)

✔ Check: x*(m² + 6mn + 8n²) = original ✔️

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Problem ⑦: -m²v - 8mvn - 12n²v

Factor out common factor: all have ‘-v’ (or just v, but let’s take negative to make leading positive)

Actually, better to factor out ‘-v’:

→ -v(m² + 8mn + 12n²)

Now factor m² + 8mn + 12n² → find two numbers that multiply to 12n² and add to 8n → 2n and 6n

→ (m + 2n)(m + 6n)

Final:
→ -v(m + 2n)(m + 6n)

✔ Check: -v*(m² + 8mn + 12n²) = original ✔️

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Problem ⑧: 7x²y - 27xy² + 18y³

Factor out common factor: all have ‘y’ → y(7x² - 27xy + 18y²)

Now factor 7x² - 27xy + 18y²

Use AC method: A=7, C=18 → 7×18=126
Need two numbers that multiply to 126 and add to -27 → -21 and -6

Split middle:
7x² -21xy -6xy + 18y²
Group: (7x² -21xy) + (-6xy + 18y²)
= 7x(x - 3y) -6y(x - 3y)
→ (7x - 6y)(x - 3y)

Final:
→ y(7x - 6y)(x - 3y)

✔ Check: y*(7x² -27xy +18y²) = original ✔️

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Problem ⑨: 4x² + 4xy - 35y²

Treat like quadratic in x: 4x² + 4xy -35y²

AC = 4 × (-35) = -140
Need two numbers that multiply to -140 and add to 4 → 14 and -10

Split: 4x² +14xy -10xy -35y²
Group: (4x² +14xy) + (-10xy -35y²)
= 2x(2x + 7y) -5y(2x + 7y)
→ (2x - 5y)(2x + 7y)

✔ Check: (2x-5y)(2x+7y) = 4x² +14xy -10xy -35y² = 4x² +4xy -35y² ✔️

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Problem ⑩: -42x² + 306xy - 84y²

First, factor out GCF. All coefficients divisible by 6? Let’s see:

-42, 306, -84 → GCF is 6? But signs: let’s factor out -6 to make leading positive.

→ -6(7x² - 51xy + 14y²)

Wait — check: -6 × 7x² = -42x² ✔️
-6 × (-51xy) = +306xy ✔️
-6 × 14y² = -84y² ✔️

Now factor 7x² -51xy +14y²

AC = 7×14 = 98
Need two numbers that multiply to 98 and add to -51 → -49 and -2

Split: 7x² -49xy -2xy +14y²
Group: (7x² -49xy) + (-2xy +14y²)
= 7x(x - 7y) -2y(x - 7y)
→ (7x - 2y)(x - 7y)

Final:
→ -6(7x - 2y)(x - 7y)

✔ Check: -6*(7x² -51xy +14y²) = original ✔️

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Final Answer:
① (x + 2)(x + 4)
② (x + 9)(x - 2)
③ (2x - 3)(3x + 4)
④ (x² + 5)(x - 3)
⑤ 2(5x - 6)(5x + 6)
⑥ x(m + 2n)(m + 4n)
⑦ -v(m + 2n)(m + 6n)
⑧ y(7x - 6y)(x - 3y)
⑨ (2x - 5y)(2x + 7y)
⑩ -6(7x - 2y)(x - 7y)