Question

factoring trinomials (a = 1)
factor each completely.

1. $b^2 + 8b + 7$
2. $n^2 - 11n + 10$
3. $m^2 + m - 90$
4. $n^2 + 4n - 12$
5. $n^2 - 10n + 9$
6. $b^2 + 16b + 64$
7. $m^2 + 2m - 24$
8. $x^2 - 4x + 24$
9. $k^2 - 13k + 40$
10. $a^2 + 11a + 18$
11. $n^2 - n - 56$
12. $n^2 - 5n + 6$

Explanation:

1) Step1: Find factors of 7 summing to 8

Factors are 1 and 7.

1) Step2: Write factored form

$(b+1)(b+7)$
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2) Step1: Find factors of 10 summing to -11

Factors are -1 and -10.

2) Step2: Write factored form

$(n-1)(n-10)$
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3) Step1: Find factors of -90 summing to 1

Factors are 10 and -9.

3) Step2: Write factored form

$(m+10)(m-9)$
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4) Step1: Find factors of -12 summing to 4

Factors are 6 and -2.

4) Step2: Write factored form

$(n+6)(n-2)$
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5) Step1: Find factors of 9 summing to -10

Factors are -1 and -9.

5) Step2: Write factored form

$(n-1)(n-9)$
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6) Step1: Identify perfect square trinomial

$b^2+16b+64 = b^2+2(8b)+8^2$

6) Step2: Write factored form

$(b+8)^2$
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7) Step1: Find factors of -24 summing to 2

Factors are 6 and -4.

7) Step2: Write factored form

$(m+6)(m-4)$
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8) Step1: Check discriminant for factorization

Discriminant: $(-4)^2-4(1)(24)=16-96=-80<0$

8) Step2: State non-factorability over reals

Trinomial is prime (cannot be factored over real numbers)
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9) Step1: Find factors of 40 summing to -13

Factors are -5 and -8.

9) Step2: Write factored form

$(k-5)(k-8)$
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10) Step1: Find factors of 18 summing to 11

Factors are 2 and 9.

10) Step2: Write factored form

$(a+2)(a+9)$
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11) Step1: Find factors of -56 summing to -1

Factors are -8 and 7.

11) Step2: Write factored form

$(n-8)(n+7)$
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12) Step1: Find factors of 6 summing to -5

Factors are -2 and -3.

12) Step2: Write factored form

$(n-2)(n-3)$

Answer:

1. $\boldsymbol{(b+1)(b+7)}$
2. $\boldsymbol{(n-1)(n-10)}$
3. $\boldsymbol{(m+10)(m-9)}$
4. $\boldsymbol{(n+6)(n-2)}$
5. $\boldsymbol{(n-1)(n-9)}$
6. $\boldsymbol{(b+8)^2}$
7. $\boldsymbol{(m+6)(m-4)}$
8. $\boldsymbol{\text{Prime (cannot be factored over real numbers)}}$
9. $\boldsymbol{(k-5)(k-8)}$
10. $\boldsymbol{(a+2)(a+9)}$
11. $\boldsymbol{(n-8)(n+7)}$
12. $\boldsymbol{(n-2)(n-3)}$