21x+20 Algebra Worksheet - Section 10.5 Name _ Factoring ... - Free Printable

Let's solve each problem step-by-step from your algebra worksheet. The goal is to factor completely the given polynomials and solve equations by factoring.

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Section: Factoring Polynomials of the form $x^2 + bx + c$ with GCFs

We'll factor each expression completely, starting with factoring out the Greatest Common Factor (GCF) if possible, then factoring the quadratic or higher-degree polynomial.

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#### 1. $2x^2 + 6x + 4$

- GCF: 2
- Factor out: $2(x^2 + 3x + 2)$
- Now factor $x^2 + 3x + 2$: find two numbers that multiply to 2 and add to 3 → 1 and 2
- So: $2(x + 1)(x + 2)$

✔ Answer: $2(x + 1)(x + 2)$

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#### 2. $4a^2 - 12a + 8$

- GCF: 4
- Factor out: $4(a^2 - 3a + 2)$
- Factor $a^2 - 3a + 2$: numbers that multiply to 2 and add to -3 → -1 and -2
- So: $4(a - 1)(a - 2)$

✔ Answer: $4(a - 1)(a - 2)$

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#### 3. $10a^2 + 10 - 20$

Wait: $10a^2 + 10 - 20 = 10a^2 - 10$

- GCF: 10
- $10(a^2 - 1)$
- $a^2 - 1$ is a difference of squares: $(a - 1)(a + 1)$

✔ Answer: $10(a - 1)(a + 1)$

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#### 4. $7a^2 - 14a - 21$

- GCF: 7
- $7(a^2 - 2a - 3)$
- Factor $a^2 - 2a - 3$: numbers that multiply to -3 and add to -2 → -3 and +1
- So: $7(a - 3)(a + 1)$

✔ Answer: $7(a - 3)(a + 1)$

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#### 5. $3y^2 - 15y + 18$

- GCF: 3
- $3(y^2 - 5y + 6)$
- Factor $y^2 - 5y + 6$: numbers that multiply to 6 and add to -5 → -2 and -3
- So: $3(y - 2)(y - 3)$

✔ Answer: $3(y - 2)(y - 3)$

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#### 6. $a^3 - 5a^2 + 4a$

- GCF: $a$
- $a(a^2 - 5a + 4)$
- Factor $a^2 - 5a + 4$: numbers that multiply to 4 and add to -5 → -4 and -1
- So: $a(a - 4)(a - 1)$

✔ Answer: $a(a - 4)(a - 1)$

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#### 7. $x^4 - 15x^3 + 56x^2$

- GCF: $x^2$
- $x^2(x^2 - 15x + 56)$
- Factor $x^2 - 15x + 56$: numbers that multiply to 56 and add to -15 → -7 and -8
- So: $x^2(x - 7)(x - 8)$

✔ Answer: $x^2(x - 7)(x - 8)$

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#### 8. $b^4 - 3b^3 - 10b^2$

- GCF: $b^2$
- $b^2(b^2 - 3b - 10)$
- Factor $b^2 - 3b - 10$: numbers that multiply to -10 and add to -3 → -5 and +2
- So: $b^2(b - 5)(b + 2)$

✔ Answer: $b^2(b - 5)(b + 2)$

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#### 9. $2a^3 + 8a^2 - 64a$

- GCF: $2a$
- $2a(a^2 + 4a - 32)$
- Factor $a^2 + 4a - 32$: numbers that multiply to -32 and add to 4 → 8 and -4
- So: $2a(a + 8)(a - 4)$

✔ Answer: $2a(a + 8)(a - 4)$

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#### 10. $3a^3 - 9a^2 - 54a$

- GCF: $3a$
- $3a(a^2 - 3a - 18)$
- Factor $a^2 - 3a - 18$: numbers that multiply to -18 and add to -3 → -6 and +3
- So: $3a(a - 6)(a + 3)$

✔ Answer: $3a(a - 6)(a + 3)$

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#### 11. $9p^2 - 54p + 72$

- GCF: 9
- $9(p^2 - 6p + 8)$
- Factor $p^2 - 6p + 8$: numbers that multiply to 8 and add to -6 → -4 and -2
- So: $9(p - 4)(p - 2)$

✔ Answer: $9(p - 4)(p - 2)$

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#### 12. $4y^3 - 4y^2 - 24y$

- GCF: $4y$
- $4y(y^2 - y - 6)$
- Factor $y^2 - y - 6$: numbers that multiply to -6 and add to -1 → -3 and +2
- So: $4y(y - 3)(y + 2)$

✔ Answer: $4y(y - 3)(y + 2)$

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#### 13. $3x^4 - 21x^3 + 10x^2$

- GCF: $x^2$
- $x^2(3x^2 - 21x + 10)$
- Now factor $3x^2 - 21x + 10$: use AC method
- AC = $3 \cdot 10 = 30$
- Find factors of 30 that add to -21 → -20 and -1? No.
Wait: try pairs: (-30,1), (-15,2), (-10,3), (-6,5) → none add to -21?
- Actually: check discriminant: $(-21)^2 - 4(3)(10) = 441 - 120 = 321$, not a perfect square → does not factor nicely over integers

But wait — maybe I made a mistake. Let's double-check:

Is $3x^2 - 21x + 10$ factorable?

Try factoring:
We need two numbers that multiply to $3 \cdot 10 = 30$, and add to -21.

- -20 and -1 → product = 20 → no
- -30 and +1 → sum = -29 → no
- -15 and -2 → sum = -17 → no
- -10 and -3 → sum = -13 → no
- -6 and -5 → sum = -11 → no

No pair adds to -21 → so it does not factor over integers.

So we leave it as is.

✔ Answer: $x^2(3x^2 - 21x + 10)$

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#### 14. $5x^4 - 10x^3 - 75x^2$

- GCF: $5x^2$
- $5x^2(x^2 - 2x - 15)$
- Factor $x^2 - 2x - 15$: numbers that multiply to -15 and add to -2 → -5 and +3
- So: $5x^2(x - 5)(x + 3)$

✔ Answer: $5x^2(x - 5)(x + 3)$

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Solve each equation by factoring

Now we solve equations set equal to zero by factoring.

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#### 15. $3x^2 + 15x + 18 = 0$

- GCF: 3
- $3(x^2 + 5x + 6) = 0$
- Factor $x^2 + 5x + 6$: numbers that multiply to 6 and add to 5 → 2 and 3
- So: $3(x + 2)(x + 3) = 0$
- Set each factor to zero:
- $x + 2 = 0$ → $x = -2$
- $x + 3 = 0$ → $x = -3$

✔ Solutions: $x = -2, -3$

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#### 16. $2x^2 + 16x + 24 = 0$

- GCF: 2
- $2(x^2 + 8x + 12) = 0$
- Factor $x^2 + 8x + 12$: numbers that multiply to 12 and add to 8 → 6 and 2
- So: $2(x + 6)(x + 2) = 0$
- Solutions:
- $x = -6$, $x = -2$

✔ Solutions: $x = -6, -2$

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#### 17. $5x^2 - 35x + 60 = 0$

- GCF: 5
- $5(x^2 - 7x + 12) = 0$
- Factor $x^2 - 7x + 12$: numbers that multiply to 12 and add to -7 → -3 and -4
- So: $5(x - 3)(x - 4) = 0$
- Solutions:
- $x = 3$, $x = 4$

✔ Solutions: $x = 3, 4$

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#### 18. $x^2 + 11x - 12 = 0$

- Already monic (leading coefficient 1)
- Find two numbers that multiply to -12 and add to 11 → 12 and -1
- So: $(x + 12)(x - 1) = 0$
- Solutions:
- $x = -12$, $x = 1$

✔ Solutions: $x = -12, 1$

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#### 19. $2y^2 + 10y = 28$

- First, move all terms to one side:
$2y^2 + 10y - 28 = 0$
- GCF: 2
- $2(y^2 + 5y - 14) = 0$
- Now factor $y^2 + 5y - 14$: need two numbers that multiply to -14 and add to 5 → 7 and -2
- So: $2(y + 7)(y - 2) = 0$
- Solutions:
- $y = -7$, $y = 2$

✔ Solutions: $y = -7, 2$

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#### 20. $6y^2 + 36 = 30y$

- Move all terms to left:
$6y^2 - 30y + 36 = 0$
- GCF: 6
- $6(y^2 - 5y + 6) = 0$
- Factor $y^2 - 5y + 6$: numbers that multiply to 6 and add to -5 → -3 and -2
- So: $6(y - 3)(y - 2) = 0$
- Solutions:
- $y = 3$, $y = 2$

✔ Solutions: $y = 2, 3$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $2(x + 1)(x + 2)$ |
| 2 | $4(a - 1)(a - 2)$ |
| 3 | $10(a - 1)(a + 1)$ |
| 4 | $7(a - 3)(a + 1)$ |
| 5 | $3(y - 2)(y - 3)$ |
| 6 | $a(a - 4)(a - 1)$ |
| 7 | $x^2(x - 7)(x - 8)$ |
| 8 | $b^2(b - 5)(b + 2)$ |
| 9 | $2a(a + 8)(a - 4)$ |
| 10 | $3a(a - 6)(a + 3)$ |
| 11 | $9(p - 4)(p - 2)$ |
| 12 | $4y(y - 3)(y + 2)$ |
| 13 | $x^2(3x^2 - 21x + 10)$ |
| 14 | $5x^2(x - 5)(x + 3)$ |
| 15 | $x = -2, -3$ |
| 16 | $x = -6, -2$ |
| 17 | $x = 3, 4$ |
| 18 | $x = -12, 1$ |
| 19 | $y = -7, 2$ |
| 20 | $y = 2, 3$ |

Let me know if you'd like explanations for any specific steps!