Factoring Polynomials Notes and Worksheets - Lindsay Bowden - Free Printable

Sure! Let's factor each of the 10 trinomials completely. Since the leading coefficient (a) is greater than 1 in all cases, we’ll use the “AC Method” (also called factoring by grouping).

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🔢 The AC Method Summary:

For a trinomial of the form `ax² + bx + c`:
1. Multiply `a × c`.
2. Find two numbers that multiply to `a×c` and add to `b`.
3. Rewrite the middle term (`bx`) using those two numbers.
4. Factor by grouping.
5. Factor out the GCF if possible.

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Let’s solve each one step-by-step.

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## ✔ 1. `4x² + 4x - 35`

- a = 4, b = 4, c = -35 → ac = 4 × (-35) = -140
- Find two numbers that multiply to -140 and add to 4 → 14 and -10

Rewrite:
> 4x² + 14x - 10x - 35

Group:
> (4x² + 14x) + (-10x - 35)

Factor each group:
> 2x(2x + 7) -5(2x + 7)

Factor out common binomial:
> (2x + 7)(2x - 5)

✔ Answer: (2x + 7)(2x - 5)

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## ✔ 2. `5x² + 37x - 24`

- a = 5, c = -24 → ac = -120
- Need two numbers that multiply to -120, add to 37 → 40 and -3

Rewrite:
> 5x² + 40x - 3x - 24

Group:
> (5x² + 40x) + (-3x - 24)

Factor:
> 5x(x + 8) -3(x + 8)

Factor out common binomial:
> (5x - 3)(x + 8)

✔ Answer: (5x - 3)(x + 8)

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## ✔ 3. `3x² + 12x + 12`

First, factor out the GCF: 3

> 3(x² + 4x + 4)

Now factor the trinomial inside:
> x² + 4x + 4 = (x + 2)²

So overall:
> 3(x + 2)²

✔ Answer: 3(x + 2)²

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## ✔ 4. `6x² - 44x + 14`

First, factor out GCF: 2

> 2(3x² - 22x + 7)

Now factor `3x² - 22x + 7`
- ac = 3 × 7 = 21
- Need two numbers that multiply to 21 and add to -22 → -21 and -1

Rewrite:
> 3x² - 21x - x + 7

Group:
> (3x² - 21x) + (-x + 7)

Factor:
> 3x(x - 7) -1(x - 7)

Factor out common binomial:
> (3x - 1)(x - 7)

Multiply back the GCF:
> 2(3x - 1)(x - 7)

✔ Answer: 2(3x - 1)(x - 7)

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## ✔ 5. `3x² - 23x - 36`

- a = 3, c = -36 → ac = -108
- Need two numbers that multiply to -108 and add to -23 → -27 and 4

Rewrite:
> 3x² - 27x + 4x - 36

Group:
> (3x² - 27x) + (4x - 36)

Factor:
> 3x(x - 9) + 4(x - 9)

Factor out common binomial:
> (3x + 4)(x - 9)

✔ Answer: (3x + 4)(x - 9)

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## ✔ 6. `2x² - 3x + 1`

- a = 2, c = 1 → ac = 2
- Need two numbers that multiply to 2 and add to -3 → -2 and -1

Rewrite:
> 2x² - 2x - x + 1

Group:
> (2x² - 2x) + (-x + 1)

Factor:
> 2x(x - 1) -1(x - 1)

Factor out common binomial:
> (2x - 1)(x - 1)

✔ Answer: (2x - 1)(x - 1)

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## ✔ 7. `20x² + 2x - 6`

First, factor out GCF: 2

> 2(10x² + x - 3)

Now factor `10x² + x - 3`
- ac = 10 × (-3) = -30
- Need two numbers that multiply to -30 and add to 1 → 6 and -5

Rewrite:
> 10x² + 6x - 5x - 3

Group:
> (10x² + 6x) + (-5x - 3)

Factor:
> 2x(5x + 3) -1(5x + 3)

Factor out common binomial:
> (2x - 1)(5x + 3)

Multiply back the GCF:
> 2(2x - 1)(5x + 3)

✔ Answer: 2(2x - 1)(5x + 3)

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## ✔ 8. `3x² + 25x + 42`

- a = 3, c = 42 → ac = 126
- Need two numbers that multiply to 126 and add to 25 → 18 and 7

Rewrite:
> 3x² + 18x + 7x + 42

Group:
> (3x² + 18x) + (7x + 42)

Factor:
> 3x(x + 6) + 7(x + 6)

Factor out common binomial:
> (3x + 7)(x + 6)

✔ Answer: (3x + 7)(x + 6)

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## ✔ 9. `2x² - 3x - 20`

- a = 2, c = -20 → ac = -40
- Need two numbers that multiply to -40 and add to -3 → -8 and 5

Rewrite:
> 2x² - 8x + 5x - 20

Group:
> (2x² - 8x) + (5x - 20)

Factor:
> 2x(x - 4) + 5(x - 4)

Factor out common binomial:
> (2x + 5)(x - 4)

✔ Answer: (2x + 5)(x - 4)

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## ✔ 10. `35x² - 11x - 6`

- a = 35, c = -6 → ac = -210
- Need two numbers that multiply to -210 and add to -11 → -21 and 10

Rewrite:
> 35x² - 21x + 10x - 6

Group:
> (35x² - 21x) + (10x - 6)

Factor:
> 7x(5x - 3) + 2(5x - 3)

Factor out common binomial:
> (7x + 2)(5x - 3)

✔ Answer: (7x + 2)(5x - 3)

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## 📝 Final Answers:

1. (2x + 7)(2x - 5)
2. (5x - 3)(x + 8)
3. 3(x + 2)²
4. 2(3x - 1)(x - 7)
5. (3x + 4)(x - 9)
6. (2x - 1)(x - 1)
7. 2(2x - 1)(5x + 3)
8. (3x + 7)(x + 6)
9. (2x + 5)(x - 4)
10. (7x + 2)(5x - 3)

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Let me know if you’d like to see any step explained in more detail! 😊