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:

Step1: Find pair for $b^2+8b+7$

Find two numbers that multiply to $7$ and add to $8$: $1$ and $7$.
$\boldsymbol{(b+1)(b+7)}$

Step2: Find pair for $n^2-11n+10$

Find two numbers that multiply to $10$ and add to $-11$: $-1$ and $-10$.
$\boldsymbol{(n-1)(n-10)}$

Step3: Find pair for $m^2+m-90$

Find two numbers that multiply to $-90$ and add to $1$: $10$ and $-9$.
$\boldsymbol{(m+10)(m-9)}$

Step4: Find pair for $n^2+4n-12$

Find two numbers that multiply to $-12$ and add to $4$: $6$ and $-2$.
$\boldsymbol{(n+6)(n-2)}$

Step5: Find pair for $n^2-10n+9$

Find two numbers that multiply to $9$ and add to $-10$: $-1$ and $-9$.
$\boldsymbol{(n-1)(n-9)}$

Step6: Find pair for $b^2+16b+64$

Find two numbers that multiply to $64$ and add to $16$: $8$ and $8$.
$\boldsymbol{(b+8)^2}$

Step7: Find pair for $m^2+2m-24$

Find two numbers that multiply to $-24$ and add to $2$: $6$ and $-4$.
$\boldsymbol{(m+6)(m-4)}$

Step8: Check $x^2-4x+24$

No real number pair multiplies to $24$ and adds to $-4$.
$\boldsymbol{\text{Prime (cannot be factored over reals)}}$

Step9: Find pair for $k^2-13k+40$

Find two numbers that multiply to $40$ and add to $-13$: $-5$ and $-8$.
$\boldsymbol{(k-5)(k-8)}$

Step10: Find pair for $a^2+11a+18$

Find two numbers that multiply to $18$ and add to $11$: $2$ and $9$.
$\boldsymbol{(a+2)(a+9)}$

Step11: Find pair for $n^2-n-56$

Find two numbers that multiply to $-56$ and add to $-1$: $-8$ and $7$.
$\boldsymbol{(n-8)(n+7)}$

Step12: Find pair for $n^2-5n+6$

Find two numbers that multiply to $6$ and add to $-5$: $-2$ and $-3$.
$\boldsymbol{(n-2)(n-3)}$

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. $\boldsymbol{\text{Prime (cannot be factored over reals)}}$
9. $\boldsymbol{(k-5)(k-8)}$
10. $\boldsymbol{(a+2)(a+9)}$
11. $\boldsymbol{(n-8)(n+7)}$
12. $\boldsymbol{(n-2)(n-3)}$