Factoring Polynomials Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each trinomial by factoring completely. We'll use the AC method or trial and error for factoring trinomials of the form $ ax^2 + bx + c $ where $ a > 1 $. The goal is to find two numbers that multiply to $ a \cdot c $ and add to $ b $, then split the middle term and factor by grouping.

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1. $ 4x^2 + 4x - 35 $

- $ a = 4 $, $ b = 4 $, $ c = -35 $
- $ a \cdot c = 4 \cdot (-35) = -140 $
- Find two numbers that multiply to $-140$ and add to $4$:
Try: $ 14 $ and $-10$: $14 \cdot (-10) = -140$, $14 + (-10) = 4$ ✔
- Split the middle term:
$$
4x^2 + 14x - 10x - 35
$$
- Group:
$$
(4x^2 + 14x) + (-10x - 35) = 2x(2x + 7) -5(2x + 7)
$$
- Factor:
$$
(2x - 5)(2x + 7)
$$

✔ Answer: $ (2x - 5)(2x + 7) $

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2. $ 5x^2 + 37x - 24 $

- $ a = 5 $, $ b = 37 $, $ c = -24 $
- $ a \cdot c = 5 \cdot (-24) = -120 $
- Find two numbers that multiply to $-120$ and add to $37$:
Try: $40$ and $-3$: $40 \cdot (-3) = -120$, $40 + (-3) = 37$ ✔
- Split:
$$
5x^2 + 40x - 3x - 24
$$
- Group:
$$
(5x^2 + 40x) + (-3x - 24) = 5x(x + 8) -3(x + 8)
$$
- Factor:
$$
(5x - 3)(x + 8)
$$

✔ Answer: $ (5x - 3)(x + 8) $

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3. $ 3x^2 + 12x + 12 $

- First, factor out GCF: $ 3 $
$$
3(x^2 + 4x + 4)
$$
- Now factor $ x^2 + 4x + 4 $: perfect square trinomial
$$
x^2 + 4x + 4 = (x + 2)^2
$$
- So:
$$
3(x + 2)^2
$$

✔ Answer: $ 3(x + 2)^2 $

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4. $ 6x^2 - 44x + 14 $

- First, factor out GCF: $ 2 $
$$
2(3x^2 - 22x + 7)
$$
- Now factor $ 3x^2 - 22x + 7 $
- $ a = 3 $, $ b = -22 $, $ c = 7 $
- $ a \cdot c = 3 \cdot 7 = 21 $
- Find two numbers that multiply to $21$ and add to $-22$:
Try: $-21$ and $-1$: $-21 \cdot (-1) = 21$, $-21 + (-1) = -22$ ✔
- Split:
$$
3x^2 - 21x - x + 7
$$
- Group:
$$
(3x^2 - 21x) + (-x + 7) = 3x(x - 7) -1(x - 7)
$$
- Factor:
$$
(3x - 1)(x - 7)
$$
- Multiply back the GCF:
$$
2(3x - 1)(x - 7)
$$

✔ Answer: $ 2(3x - 1)(x - 7) $

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5. $ 3x^2 - 23x - 36 $

- $ a = 3 $, $ b = -23 $, $ c = -36 $
- $ a \cdot c = 3 \cdot (-36) = -108 $
- Find two numbers that multiply to $-108$ and add to $-23$:
Try: $-27$ and $4$: $-27 \cdot 4 = -108$, $-27 + 4 = -23$ ✔
- Split:
$$
3x^2 - 27x + 4x - 36
$$
- Group:
$$
(3x^2 - 27x) + (4x - 36) = 3x(x - 9) + 4(x - 9)
$$
- Factor:
$$
(3x + 4)(x - 9)
$$

✔ Answer: $ (3x + 4)(x - 9) $

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6. $ 2x^2 - 3x + 1 $

- $ a = 2 $, $ b = -3 $, $ c = 1 $
- $ a \cdot c = 2 \cdot 1 = 2 $
- Find two numbers that multiply to $2$ and add to $-3$:
Try: $-2$ and $-1$: $-2 \cdot (-1) = 2$, $-2 + (-1) = -3$ ✔
- Split:
$$
2x^2 - 2x - x + 1
$$
- Group:
$$
(2x^2 - 2x) + (-x + 1) = 2x(x - 1) -1(x - 1)
$$
- Factor:
$$
(2x - 1)(x - 1)
$$

✔ Answer: $ (2x - 1)(x - 1) $

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7. $ 20x^2 + 2x - 6 $

- First, factor out GCF: $ 2 $
$$
2(10x^2 + x - 3)
$$
- Now factor $ 10x^2 + x - 3 $
- $ a = 10 $, $ b = 1 $, $ c = -3 $
- $ a \cdot c = 10 \cdot (-3) = -30 $
- Find two numbers that multiply to $-30$ and add to $1$:
Try: $6$ and $-5$: $6 \cdot (-5) = -30$, $6 + (-5) = 1$ ✔
- Split:
$$
10x^2 + 6x - 5x - 3
$$
- Group:
$$
(10x^2 + 6x) + (-5x - 3) = 2x(5x + 3) -1(5x + 3)
$$
- Factor:
$$
(2x - 1)(5x + 3)
$$
- Multiply back GCF:
$$
2(2x - 1)(5x + 3)
$$

✔ Answer: $ 2(2x - 1)(5x + 3) $

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8. $ 3x^2 + 25x + 42 $

- $ a = 3 $, $ b = 25 $, $ c = 42 $
- $ a \cdot c = 3 \cdot 42 = 126 $
- Find two numbers that multiply to $126$ and add to $25$:
Try: $18$ and $7$: $18 \cdot 7 = 126$, $18 + 7 = 25$ ✔
- Split:
$$
3x^2 + 18x + 7x + 42
$$
- Group:
$$
(3x^2 + 18x) + (7x + 42) = 3x(x + 6) + 7(x + 6)
$$
- Factor:
$$
(3x + 7)(x + 6)
$$

✔ Answer: $ (3x + 7)(x + 6) $

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9. $ 2x^2 - 3x - 20 $

- $ a = 2 $, $ b = -3 $, $ c = -20 $
- $ a \cdot c = 2 \cdot (-20) = -40 $
- Find two numbers that multiply to $-40$ and add to $-3$:
Try: $-8$ and $5$: $-8 \cdot 5 = -40$, $-8 + 5 = -3$ ✔
- Split:
$$
2x^2 - 8x + 5x - 20
$$
- Group:
$$
(2x^2 - 8x) + (5x - 20) = 2x(x - 4) + 5(x - 4)
$$
- Factor:
$$
(2x + 5)(x - 4)
$$

✔ Answer: $ (2x + 5)(x - 4) $

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10. $ 35x^2 - 11x - 6 $

- $ a = 35 $, $ b = -11 $, $ c = -6 $
- $ a \cdot c = 35 \cdot (-6) = -210 $
- Find two numbers that multiply to $-210$ and add to $-11$:
Try: $-21$ and $10$: $-21 \cdot 10 = -210$, $-21 + 10 = -11$ ✔
- Split:
$$
35x^2 - 21x + 10x - 6
$$
- Group:
$$
(35x^2 - 21x) + (10x - 6) = 7x(5x - 3) + 2(5x - 3)
$$
- Factor:
$$
(7x + 2)(5x - 3)
$$

✔ Answer: $ (7x + 2)(5x - 3) $

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✔ Final Answers:

| Problem | Factored Form |
|--------|----------------|
| 1 | $ (2x - 5)(2x + 7) $ |
| 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) $ |

Let me know if you'd like these explained step-by-step with diagrams or a visual method!