I'll solve each of these factoring problems step by step.
Problem 1: 4x² + 4x - 35
- Need two numbers that multiply to (4)(-35) = -140 and add to 4
- Those numbers are 14 and -10
- Rewrite: 4x² + 14x - 10x - 35
- Factor by grouping: 2x(2x + 7) - 5(2x + 7)
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Answer: (2x + 7)(2x - 5)
Problem 2: 5x² + 37x - 24
- Need two numbers that multiply to (5)(-24) = -120 and add to 37
- Those numbers are 40 and -3
- Rewrite: 5x² + 40x - 3x - 24
- Factor by grouping: 5x(x + 8) - 3(x + 8)
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Answer: (x + 8)(5x - 3)
Problem 3: 3x² + 12x + 12
- First factor out GCF: 3(x² + 4x + 4)
- x² + 4x + 4 is a perfect square trinomial
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Answer: 3(x + 2)²
Problem 4: 6x² - 44x + 14
- First factor out GCF: 2(3x² - 22x + 7)
- Need two numbers that multiply to (3)(7) = 21 and add to -22
- Those numbers are -21 and -1
- Rewrite inside: 3x² - 21x - x + 7
- Factor by grouping: 3x(x - 7) - 1(x - 7)
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Answer: 2(3x - 1)(x - 7)
Problem 5: 3x² - 23x - 36
- Need two numbers that multiply to (3)(-36) = -108 and add to -23
- Those numbers are -27 and 4
- Rewrite: 3x² - 27x + 4x - 36
- Factor by grouping: 3x(x - 9) + 4(x - 9)
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Answer: (x - 9)(3x + 4)
Problem 6: 2x² - 3x + 1
- Need two numbers that multiply to (2)(1) = 2 and add to -3
- Those numbers are -2 and -1
- Rewrite: 2x² - 2x - x + 1
- Factor by grouping: 2x(x - 1) - 1(x - 1)
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Answer: (x - 1)(2x - 1)
Problem 7: 20x² + 2x - 6
- First factor out GCF: 2(10x² + x - 3)
- Need two numbers that multiply to (10)(-3) = -30 and add to 1
- Those numbers are 6 and -5
- Rewrite inside: 10x² + 6x - 5x - 3
- Factor by grouping: 2x(5x + 3) - 1(5x + 3)
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Answer: 2(2x - 1)(5x + 3)
Problem 8: 3x² + 25x + 42
- Need two numbers that multiply to (3)(42) = 126 and add to 25
- Those numbers are 7 and 18
- Rewrite: 3x² + 18x + 7x + 42
- Factor by grouping: 3x(x + 6) + 7(x + 6)
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Answer: (x + 6)(3x + 7)
Problem 9: 2x² - 3x - 20
- Need two numbers that multiply to (2)(-20) = -40 and add to -3
- Those numbers are -8 and 5
- Rewrite: 2x² - 8x + 5x - 20
- Factor by grouping: 2x(x - 4) + 5(x - 4)
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Answer: (x - 4)(2x + 5)
Problem 10: 35x² - 11x - 6
- Need two numbers that multiply to (35)(-6) = -210 and add to -11
- Those numbers are -21 and 10
- Rewrite: 35x² - 21x + 10x - 6
- Factor by grouping: 7x(5x - 3) + 2(5x - 3)
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Answer: (5x - 3)(7x + 2)
Final Answer:
1. (2x + 7)(2x - 5)
2. (x + 8)(5x - 3)
3. 3(x + 2)²
4. 2(3x - 1)(x - 7)
5. (x - 9)(3x + 4)
6. (x - 1)(2x - 1)
7. 2(2x - 1)(5x + 3)
8. (x + 6)(3x + 7)
9. (x - 4)(2x + 5)
10. (5x - 3)(7x + 2)
Parent Tip: Review the logic above to help your child master the concept of factoring polynomials by grouping worksheet.
