Algebra worksheet for factoring trinomials, Lesson 7-3, Practice C.

Let’s solve each trinomial by factoring. We’re looking for two binomials that multiply to give the original trinomial.

For a trinomial like \( x^2 + bx + c \), we need two numbers that:
- Multiply to \( c \) (the constant term)
- Add up to \( b \) (the coefficient of the middle term)

We’ll go one by one.

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1. \( x^2 + 10x + 24 \)
Find two numbers that multiply to 24 and add to 10 → 6 and 4
→ \( (x + 6)(x + 4) \)

2. \( y^2 + 12y + 20 \)
Multiply to 20, add to 12 → 10 and 2
→ \( (y + 10)(y + 2) \)

3. \( a^2 + 15a + 54 \)
Multiply to 54, add to 15 → 9 and 6
→ \( (a + 9)(a + 6) \)

4. \( h^2 + 18h + 45 \)
Multiply to 45, add to 18 → 15 and 3
→ \( (h + 15)(h + 3) \)

5. \( x^2 + 16x + 48 \)
Multiply to 48, add to 16 → 12 and 4
→ \( (x + 12)(x + 4) \)

6. \( c^2 + 15c + 50 \)
Multiply to 50, add to 15 → 10 and 5
→ \( (c + 10)(c + 5) \)

7. \( x^2 - 16x + 48 \)
Multiply to 48, add to -16 → -12 and -4
→ \( (x - 12)(x - 4) \)

8. \( d^2 - 19d + 88 \)
Multiply to 88, add to -19 → -11 and -8
→ \( (d - 11)(d - 8) \)

9. \( x^2 - 20x + 36 \)
Multiply to 36, add to -20 → -18 and -2
→ \( (x - 18)(x - 2) \)

10. \( m^2 - 43m + 42 \)
Multiply to 42, add to -43 → -42 and -1
→ \( (m - 42)(m - 1) \)

11. \( x^2 - 16x + 28 \)
Multiply to 28, add to -16 → -14 and -2
→ \( (x - 14)(x - 2) \)

12. \( n^2 - 12n + 35 \)
Multiply to 35, add to -12 → -7 and -5
→ \( (n - 7)(n - 5) \)

13. \( t^2 + 3t - 28 \)
Multiply to -28, add to 3 → 7 and -4
→ \( (t + 7)(t - 4) \)

14. \( b^2 + 11b - 42 \)
Multiply to -42, add to 11 → 14 and -3
→ \( (b + 14)(b - 3) \)

15. \( x^2 + 12x - 160 \)
Multiply to -160, add to 12 → 20 and -8
→ \( (x + 20)(x - 8) \)

16. \( g^2 + 2g - 48 \)
Multiply to -48, add to 2 → 8 and -6
→ \( (g + 8)(g - 6) \)

17. \( k^2 + 16k - 36 \)
Multiply to -36, add to 16 → 18 and -2
→ \( (k + 18)(k - 2) \)

18. \( x^2 + 2x - 63 \)
Multiply to -63, add to 2 → 9 and -7
→ \( (x + 9)(x - 7) \)

19. \( p^2 - 2p - 8 \)
Multiply to -8, add to -2 → -4 and 2
→ \( (p - 4)(p + 2) \)

20. \( x^2 - x - 72 \)
Multiply to -72, add to -1 → -9 and 8
→ \( (x - 9)(x + 8) \)

21. \( q^2 - 3q - 18 \)
Multiply to -18, add to -3 → -6 and 3
→ \( (q - 6)(q + 3) \)

22. \( x^2 - 4x - 32 \)
Multiply to -32, add to -4 → -8 and 4
→ \( (x - 8)(x + 4) \)

23. \( t^2 - 10t - 39 \)
Multiply to -39, add to -10 → -13 and 3
→ \( (t - 13)(t + 3) \)

24. \( w^2 - 20w - 125 \)
Multiply to -125, add to -20 → -25 and 5
→ \( (w - 25)(w + 5) \)

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25. Factor \( n^2 + 8n - 48 \). Show that the original polynomial and the factored form describe the same sequence of numbers for \( n = 0, 1, 2, 3, 4 \).

First, factor \( n^2 + 8n - 48 \):

Look for two numbers that multiply to -48 and add to 8 → 12 and -4
→ \( (n + 12)(n - 4) \)

Now, let’s make a table for both forms:

Original: \( n^2 + 8n - 48 \)
Factored: \( (n + 12)(n - 4) \)

Test for n = 0, 1, 2, 3, 4:

| n | Original: \( n^2 + 8n - 48 \) | Factored: \( (n+12)(n-4) \) |
|---|-------------------------------|-----------------------------|
| 0 | \( 0 + 0 - 48 = -48 \) | \( (12)(-4) = -48 \) |
| 1 | \( 1 + 8 - 48 = -39 \) | \( (13)(-3) = -39 \) |
| 2 | \( 4 + 16 - 48 = -28 \) | \( (14)(-2) = -28 \) |
| 3 | \( 9 + 24 - 48 = -15 \) | \( (15)(-1) = -15 \) |
| 4 | \( 16 + 32 - 48 = 0 \) | \( (16)(0) = 0 \) |

They match! So the factored form is correct and gives the same values.

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Final Answer:

1. \( (x + 6)(x + 4) \)
2. \( (y + 10)(y + 2) \)
3. \( (a + 9)(a + 6) \)
4. \( (h + 15)(h + 3) \)
5. \( (x + 12)(x + 4) \)
6. \( (c + 10)(c + 5) \)
7. \( (x - 12)(x - 4) \)
8. \( (d - 11)(d - 8) \)
9. \( (x - 18)(x - 2) \)
10. \( (m - 42)(m - 1) \)
11. \( (x - 14)(x - 2) \)
12. \( (n - 7)(n - 5) \)
13. \( (t + 7)(t - 4) \)
14. \( (b + 14)(b - 3) \)
15. \( (x + 20)(x - 8) \)
16. \( (g + 8)(g - 6) \)
17. \( (k + 18)(k - 2) \)
18. \( (x + 9)(x - 7) \)
19. \( (p - 4)(p + 2) \)
20. \( (x - 9)(x + 8) \)
21. \( (q - 6)(q + 3) \)
22. \( (x - 8)(x + 4) \)
23. \( (t - 13)(t + 3) \)
24. \( (w - 25)(w + 5) \)
25. Factored form: \( (n + 12)(n - 4) \); tables show matching values for n=0 to 4.