Factoring using the distributive property worksheet with answer key included.

Let’s solve each problem step by step. We’re factoring using the distributive property, which means we look for the greatest common factor (GCF) of all terms in the expression and pull it out front.

We’ll go one by one:

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1. 16x² – 32x

- GCF of 16 and 32 is 16
- Both have at least one x
→ So, GCF = 16x

Factor it out:
= 16x(x) – 16x(2)
= 16x(x – 2)

✔ Final Answer: 16x(x – 2)

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2. 70b – 35

- GCF of 70 and 35 is 35
- Only first term has b → so no variable in GCF

Factor out 35:
= 35(2b) – 35(1)
= 35(2b – 1)

✔ Final Answer: 35(2b – 1)

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3. 4x – 60

- GCF of 4 and 60 is 4
- Second term has no x → so GCF is just 4

Factor out 4:
= 4(x) – 4(15)
= 4(x – 15)

✔ Final Answer: 4(x – 15)

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4. 48a² + 8a

- GCF of 48 and 8 is 8
- Both have at least one a
→ GCF = 8a

Factor out 8a:
= 8a(6a) + 8a(1)
= 8a(6a + 1)

✔ Final Answer: 8a(6a + 1)

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5. 66y + 44

- GCF of 66 and 44 is 22
- No common variable (second term has no y)

Factor out 22:
= 22(3y) + 22(2)
= 22(3y + 2)

✔ Final Answer: 22(3y + 2)

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6. 6z – 3

- GCF of 6 and 3 is 3
- Second term has no z → GCF = 3

Factor out 3:
= 3(2z) – 3(1)
= 3(2z – 1)

✔ Final Answer: 3(2z – 1)

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7. 90k² – 40k

- GCF of 90 and 40 is 10
- Both have at least one k
→ GCF = 10k

Factor out 10k:
= 10k(9k) – 10k(4)
= 10k(9k – 4)

✔ Final Answer: 10k(9k – 4)

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8. 19m + 38

- GCF of 19 and 38 is 19
- Second term has no m → GCF = 19

Factor out 19:
= 19(m) + 19(2)
= 19(m + 2)

✔ Final Answer: 19(m + 2)

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Now let’s do the second column:

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9. 22x³ – 24x² + 88x

Look at coefficients: 22, 24, 88 → GCF?

- 22 = 2×11
- 24 = 2×12
- 88 = 2×44 → GCF of numbers is 2

Variables: all have at least x → lowest power is x¹

→ GCF = 2x

Factor out 2x:

= 2x(11x²) – 2x(12x) + 2x(44)
Wait — check: 2x × 11x² = 22x³ ✔️
2x × 12x = 24x² ✔️
2x × 44 = 88x ✔️

But signs: original is –24x², so inside should be –12x

So:

= 2x(11x² – 12x + 44)

✔ Final Answer: 2x(11x² – 12x + 44)

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10. 90y⁶ + 65y³ + 95y²

Coefficients: 90, 65, 95

Find GCF:

- 90 = 5×18
- 65 = 5×13
- 95 = 5×19 → GCF = 5

Variables: powers are y⁶, y³, y² → lowest is y²

→ GCF = 5y²

Factor out 5y²:

= 5y²(18y⁴) + 5y²(13y) + 5y²(19)
Check: 5y² × 18y⁴ = 90y⁶ ✔️
5y² × 13y = 65y³ ✔️
5y² × 19 = 95y² ✔️

So:

= 5y²(18y⁴ + 13y + 19)

✔ Final Answer: 5y²(18y⁴ + 13y + 19)

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11. 81a⁹ + 15a⁵ + 63a³

Coefficients: 81, 15, 63

GCF?

- 81 = 3×27
- 15 = 3×5
- 63 = 3×21 → GCF = 3

Variables: a⁹, a⁵, a³ → lowest power is a³

→ GCF = 3a³

Factor out 3a³:

= 3a³(27a⁶) + 3a³(5a²) + 3a³(21)
Check: 3a³ × 27a⁶ = 81a⁹ ✔️
3a³ × 5a² = 15a⁵ ✔️
3a³ × 21 = 63a³ ✔️

So:

= 3a³(27a⁶ + 5a² + 21)

✔ Final Answer: 3a³(27a⁶ + 5a² + 21)

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12. 10m⁹ – 8m⁶ – 6m⁴

Coefficients: 10, 8, 6 → GCF = 2

Variables: m⁹, m⁶, m⁴ → lowest power = m⁴

→ GCF = 2m⁴

Factor out 2m⁴:

= 2m⁴(5m⁵) – 2m⁴(4m²) – 2m⁴(3)
Check: 2m⁴ × 5m⁵ = 10m⁹ ✔️
2m⁴ × 4m² = 8m⁶ ✔️
2m⁴ × 3 = 6m⁴ ✔️

Signs match original: minus, minus

So:

= 2m⁴(5m⁵ – 4m² – 3)

✔ Final Answer: 2m⁴(5m⁵ – 4m² – 3)

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13. –14x⁶ – 28x⁴ – 84x³

All terms negative → we can factor out a negative GCF to make inside positive (optional but often preferred).

Coefficients: 14, 28, 84 → GCF = 14

Variables: x⁶, x⁴, x³ → lowest = x³

Since all terms are negative, let’s factor out –14x³

Then:

= –14x³(x³) –14x³(2x) –14x³(6)
Wait — check:

–14x³ × x³ = –14x⁶ ✔️
–14x³ × 2x = –28x⁴ ✔️
–14x³ × 6 = –84x³ ✔️

So:

= –14x³(x³ + 2x + 6)

✔ Final Answer: –14x³(x³ + 2x + 6)

*(Note: You could also factor out 14x³ and leave negatives inside, but factoring out negative makes inner expression cleaner.)*

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14. 13a¹⁰ – 39a⁸ + 52a⁵

Coefficients: 13, 39, 52

13 is prime → 13 divides 39 (3×13), 52 (4×13) → GCF = 13

Variables: a¹⁰, a⁸, a⁵ → lowest = a⁵

→ GCF = 13a⁵

Factor out 13a⁵:

= 13a⁵(a⁵) – 13a⁵(3a³) + 13a⁵(4)
Check: 13a⁵ × a⁵ = 13a¹⁰ ✔️
13a⁵ × 3a³ = 39a⁸ ✔️
13a⁵ × 4 = 52a⁵ ✔️

Signs: original is –, + → matches

So:

= 13a⁵(a⁵ – 3a³ + 4)

✔ Final Answer: 13a⁵(a⁵ – 3a³ + 4)

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15. 75z⁶ + 125z⁴ + 25z²

Coefficients: 75, 125, 25 → GCF = 25

Variables: z⁶, z⁴, z² → lowest = z²

→ GCF = 25z²

Factor out 25z²:

= 25z²(3z⁴) + 25z²(5z²) + 25z²(1)
Check: 25z² × 3z⁴ = 75z⁶ ✔️
25z² × 5z² = 125z⁴ ✔️
25z² × 1 = 25z² ✔️

So:

= 25z²(3z⁴ + 5z² + 1)

✔ Final Answer: 25z²(3z⁴ + 5z² + 1)

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16. 7x⁹ – 56x⁸ + 14x⁷

Coefficients: 7, 56, 14 → GCF = 7

Variables: x⁹, x⁸, x⁷ → lowest = x⁷

→ GCF = 7x⁷

Factor out 7x⁷:

= 7x⁷(x²) – 7x⁷(8x) + 7x⁷(2)
Check: 7x⁷ × x² = 7x⁹ ✔️
7x⁷ × 8x = 56x⁸ ✔️
7x⁷ × 2 = 14x⁷ ✔️

Signs: –, + → matches

So:

= 7x⁷(x² – 8x + 2)

✔ Final Answer: 7x⁷(x² – 8x + 2)

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Final Answers List:

1. 16x(x – 2)
2. 35(2b – 1)
3. 4(x – 15)
4. 8a(6a + 1)
5. 22(3y + 2)
6. 3(2z – 1)
7. 10k(9k – 4)
8. 19(m + 2)
9. 2x(11x² – 12x + 44)
10. 5y²(18y⁴ + 13y + 19)
11. 3a³(27a⁶ + 5a² + 21)
12. 2m⁴(5m⁵ – 4m² – 3)
13. –14x³(x³ + 2x + 6)
14. 13a⁵(a⁵ – 3a³ + 4)
15. 25z²(3z⁴ + 5z² + 1)
16. 7x⁷(x² – 8x + 2)

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Final Answer:
1. 16x(x – 2)
2. 35(2b – 1)
3. 4(x – 15)
4. 8a(6a + 1)
5. 22(3y + 2)
6. 3(2z – 1)
7. 10k(9k – 4)
8. 19(m + 2)
9. 2x(11x² – 12x + 44)
10. 5y²(18y⁴ + 13y + 19)
11. 3a³(27a⁶ + 5a² + 21)
12. 2m⁴(5m⁵ – 4m² – 3)
13. –14x³(x³ + 2x + 6)
14. 13a⁵(a⁵ – 3a³ + 4)
15. 25z²(3z⁴ + 5z² + 1)
16. 7x⁷(x² – 8x + 2)