Factoring Worksheets - Math Monks - Free Printable
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Step-by-step solution for: Factoring Worksheets - Math Monks
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Show Answer Key & Explanations
Step-by-step solution for: Factoring Worksheets - Math Monks
Let’s solve each quadratic expression by factoring. We’ll go one at a time, step by step.
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1. 15x² + 141x – 90
First, look for a common factor in all terms.
All coefficients (15, 141, 90) are divisible by 3.
Factor out 3:
→ 3(5x² + 47x – 30)
Now factor 5x² + 47x – 30.
We need two numbers that multiply to (5 × -30 = -150) and add to 47.
Try pairs of factors of -150:
- 50 and -3 → 50 + (-3) = 47 ✔
Split the middle term using 50 and -3:
5x² + 50x – 3x – 30
Group:
(5x² + 50x) + (-3x – 30)
= 5x(x + 10) – 3(x + 10)
= (5x – 3)(x + 10)
Don’t forget the 3 we factored out earlier!
Final answer: 3(5x – 3)(x + 10)
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2. 16x² – 54x + 35
Check for common factor? 16, 54, 35 — no common factor other than 1.
Multiply first and last: 16 × 35 = 560
Need two numbers that multiply to 560 and add to -54.
Since sum is negative and product positive, both numbers are negative.
Try: -40 and -14 → (-40)×(-14)=560, (-40)+(-14)=-54 ✔
Split middle term:
16x² – 40x – 14x + 35
Group:
(16x² – 40x) + (-14x + 35)
= 8x(2x – 5) –7(2x – 5)
= (8x – 7)(2x – 5)
Final answer: (8x – 7)(2x – 5)
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3. 24x² + 18x – 15
Common factor? All divisible by 3.
→ 3(8x² + 6x – 5)
Factor 8x² + 6x – 5.
Multiply 8 × -5 = -40
Find two numbers that multiply to -40 and add to 6 → 10 and -4
Split:
8x² + 10x – 4x – 5
Group:
(8x² + 10x) + (-4x – 5)
= 2x(4x + 5) –1(4x + 5)
= (2x – 1)(4x + 5)
Include the 3:
Final answer: 3(2x – 1)(4x + 5)
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4. 2x² + 27x + 81
No common factor.
Multiply 2 × 81 = 162
Need two numbers that multiply to 162 and add to 27 → 18 and 9
Split:
2x² + 18x + 9x + 81
Group:
(2x² + 18x) + (9x + 81)
= 2x(x + 9) + 9(x + 9)
= (2x + 9)(x + 9)
Final answer: (2x + 9)(x + 9)
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5. 8x² – 2x – 1
No common factor.
Multiply 8 × -1 = -8
Need two numbers that multiply to -8 and add to -2 → -4 and 2
Split:
8x² – 4x + 2x – 1
Group:
(8x² – 4x) + (2x – 1)
= 4x(2x – 1) + 1(2x – 1)
= (4x + 1)(2x – 1)
Final answer: (4x + 1)(2x – 1)
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6. 3x² + 6x – 9
Common factor? Yes, 3
→ 3(x² + 2x – 3)
Factor x² + 2x – 3 → find two numbers that multiply to -3 and add to 2 → 3 and -1
→ (x + 3)(x – 1)
Include the 3:
Final answer: 3(x + 3)(x – 1)
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7. 30x² – 27x – 21
Common factor? All divisible by 3
→ 3(10x² – 9x – 7)
Factor 10x² – 9x – 7
Multiply 10 × -7 = -70
Need two numbers that multiply to -70 and add to -9 → -14 and 5
Split:
10x² – 14x + 5x – 7
Group:
(10x² – 14x) + (5x – 7)
= 2x(5x – 7) + 1(5x – 7)
= (2x + 1)(5x – 7)
Include the 3:
Final answer: 3(2x + 1)(5x – 7)
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8. 40x² + 126x + 44
Common factor? All even → divide by 2
→ 2(20x² + 63x + 22)
Factor 20x² + 63x + 22
Multiply 20 × 22 = 440
Need two numbers that multiply to 440 and add to 63 → 55 and 8
Split:
20x² + 55x + 8x + 22
Group:
(20x² + 55x) + (8x + 22)
= 5x(4x + 11) + 2(4x + 11)
= (5x + 2)(4x + 11)
Include the 2:
Final answer: 2(5x + 2)(4x + 11)
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9. 5x² + 11x + 6
No common factor.
Multiply 5 × 6 = 30
Need two numbers that multiply to 30 and add to 11 → 5 and 6
Split:
5x² + 5x + 6x + 6
Group:
(5x² + 5x) + (6x + 6)
= 5x(x + 1) + 6(x + 1)
= (5x + 6)(x + 1)
Final answer: (5x + 6)(x + 1)
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10. 7x² – 9x – 10
No common factor.
Multiply 7 × -10 = -70
Need two numbers that multiply to -70 and add to -9 → -14 and 5
Split:
7x² – 14x + 5x – 10
Group:
(7x² – 14x) + (5x – 10)
= 7x(x – 2) + 5(x – 2)
= (7x + 5)(x – 2)
Final answer: (7x + 5)(x – 2)
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Final Answer:
1. 3(5x – 3)(x + 10)
2. (8x – 7)(2x – 5)
3. 3(2x – 1)(4x + 5)
4. (2x + 9)(x + 9)
5. (4x + 1)(2x – 1)
6. 3(x + 3)(x – 1)
7. 3(2x + 1)(5x – 7)
8. 2(5x + 2)(4x + 11)
9. (5x + 6)(x + 1)
10. (7x + 5)(x – 2)
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1. 15x² + 141x – 90
First, look for a common factor in all terms.
All coefficients (15, 141, 90) are divisible by 3.
Factor out 3:
→ 3(5x² + 47x – 30)
Now factor 5x² + 47x – 30.
We need two numbers that multiply to (5 × -30 = -150) and add to 47.
Try pairs of factors of -150:
- 50 and -3 → 50 + (-3) = 47 ✔
Split the middle term using 50 and -3:
5x² + 50x – 3x – 30
Group:
(5x² + 50x) + (-3x – 30)
= 5x(x + 10) – 3(x + 10)
= (5x – 3)(x + 10)
Don’t forget the 3 we factored out earlier!
Final answer: 3(5x – 3)(x + 10)
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2. 16x² – 54x + 35
Check for common factor? 16, 54, 35 — no common factor other than 1.
Multiply first and last: 16 × 35 = 560
Need two numbers that multiply to 560 and add to -54.
Since sum is negative and product positive, both numbers are negative.
Try: -40 and -14 → (-40)×(-14)=560, (-40)+(-14)=-54 ✔
Split middle term:
16x² – 40x – 14x + 35
Group:
(16x² – 40x) + (-14x + 35)
= 8x(2x – 5) –7(2x – 5)
= (8x – 7)(2x – 5)
Final answer: (8x – 7)(2x – 5)
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3. 24x² + 18x – 15
Common factor? All divisible by 3.
→ 3(8x² + 6x – 5)
Factor 8x² + 6x – 5.
Multiply 8 × -5 = -40
Find two numbers that multiply to -40 and add to 6 → 10 and -4
Split:
8x² + 10x – 4x – 5
Group:
(8x² + 10x) + (-4x – 5)
= 2x(4x + 5) –1(4x + 5)
= (2x – 1)(4x + 5)
Include the 3:
Final answer: 3(2x – 1)(4x + 5)
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4. 2x² + 27x + 81
No common factor.
Multiply 2 × 81 = 162
Need two numbers that multiply to 162 and add to 27 → 18 and 9
Split:
2x² + 18x + 9x + 81
Group:
(2x² + 18x) + (9x + 81)
= 2x(x + 9) + 9(x + 9)
= (2x + 9)(x + 9)
Final answer: (2x + 9)(x + 9)
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5. 8x² – 2x – 1
No common factor.
Multiply 8 × -1 = -8
Need two numbers that multiply to -8 and add to -2 → -4 and 2
Split:
8x² – 4x + 2x – 1
Group:
(8x² – 4x) + (2x – 1)
= 4x(2x – 1) + 1(2x – 1)
= (4x + 1)(2x – 1)
Final answer: (4x + 1)(2x – 1)
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6. 3x² + 6x – 9
Common factor? Yes, 3
→ 3(x² + 2x – 3)
Factor x² + 2x – 3 → find two numbers that multiply to -3 and add to 2 → 3 and -1
→ (x + 3)(x – 1)
Include the 3:
Final answer: 3(x + 3)(x – 1)
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7. 30x² – 27x – 21
Common factor? All divisible by 3
→ 3(10x² – 9x – 7)
Factor 10x² – 9x – 7
Multiply 10 × -7 = -70
Need two numbers that multiply to -70 and add to -9 → -14 and 5
Split:
10x² – 14x + 5x – 7
Group:
(10x² – 14x) + (5x – 7)
= 2x(5x – 7) + 1(5x – 7)
= (2x + 1)(5x – 7)
Include the 3:
Final answer: 3(2x + 1)(5x – 7)
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8. 40x² + 126x + 44
Common factor? All even → divide by 2
→ 2(20x² + 63x + 22)
Factor 20x² + 63x + 22
Multiply 20 × 22 = 440
Need two numbers that multiply to 440 and add to 63 → 55 and 8
Split:
20x² + 55x + 8x + 22
Group:
(20x² + 55x) + (8x + 22)
= 5x(4x + 11) + 2(4x + 11)
= (5x + 2)(4x + 11)
Include the 2:
Final answer: 2(5x + 2)(4x + 11)
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9. 5x² + 11x + 6
No common factor.
Multiply 5 × 6 = 30
Need two numbers that multiply to 30 and add to 11 → 5 and 6
Split:
5x² + 5x + 6x + 6
Group:
(5x² + 5x) + (6x + 6)
= 5x(x + 1) + 6(x + 1)
= (5x + 6)(x + 1)
Final answer: (5x + 6)(x + 1)
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10. 7x² – 9x – 10
No common factor.
Multiply 7 × -10 = -70
Need two numbers that multiply to -70 and add to -9 → -14 and 5
Split:
7x² – 14x + 5x – 10
Group:
(7x² – 14x) + (5x – 10)
= 7x(x – 2) + 5(x – 2)
= (7x + 5)(x – 2)
Final answer: (7x + 5)(x – 2)
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Final Answer:
1. 3(5x – 3)(x + 10)
2. (8x – 7)(2x – 5)
3. 3(2x – 1)(4x + 5)
4. (2x + 9)(x + 9)
5. (4x + 1)(2x – 1)
6. 3(x + 3)(x – 1)
7. 3(2x + 1)(5x – 7)
8. 2(5x + 2)(4x + 11)
9. (5x + 6)(x + 1)
10. (7x + 5)(x – 2)
Parent Tip: Review the logic above to help your child master the concept of factoring expressions worksheet.
