factor each polynomial.
1. t² + 8t + 12
2. n² + 7n + 12
3. p² + 9p + 20
4. h² + 9h + 18
5. n² + 3n - 18
6. x² + 2x - 8
7. y² - 5y - 6
8. g² + 3g - 10
9. r² + 4r - 12
10. x² - x - 12
11. w² - w - 6
12. y² - 6y + 8
13. x² - 8x + 15
14. b² - 9b + 8

Question

Accepted Answer

1. **For $t^{2}+8t + 12$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to 12 and add up to 8**
The numbers are 2 and 6 since $2\times6 = 12$ and $2 + 6=8$.
     - **Step2: Rewrite the middle term and factor by grouping**
$t^{2}+8t + 12=t^{2}+2t+6t + 12=t(t + 2)+6(t + 2)=(t + 2)(t + 6)$
2. **For $n^{2}+7n + 12$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to 12 and add up to 7**
The numbers are 3 and 4 since $3\times4 = 12$ and $3+4 = 7$.
     - **Step2: Rewrite the middle term and factor by grouping**
$n^{2}+7n + 12=n^{2}+3n+4n + 12=n(n + 3)+4(n + 3)=(n + 3)(n + 4)$
3. **For $p^{2}+9p + 20$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to 20 and add up to 9**
The numbers are 4 and 5 since $4\times5 = 20$ and $4 + 5=9$.
     - **Step2: Rewrite the middle term and factor by grouping**
$p^{2}+9p + 20=p^{2}+4p+5p + 20=p(p + 4)+5(p + 4)=(p + 4)(p + 5)$
4. **For $h^{2}+9h + 18$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to 18 and add up to 9**
The numbers are 3 and 6 since $3\times6 = 18$ and $3+6 = 9$.
     - **Step2: Rewrite the middle term and factor by grouping**
$h^{2}+9h + 18=h^{2}+3h+6h + 18=h(h + 3)+6(h + 3)=(h + 3)(h + 6)$
5. **For $n^{2}+3n-18$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to - 18 and add up to 3**
The numbers are 6 and - 3 since $6\times(-3)=-18$ and $6+( - 3)=3$.
     - **Step2: Rewrite the middle term and factor by grouping**
$n^{2}+3n-18=n^{2}+6n-3n-18=n(n + 6)-3(n + 6)=(n + 6)(n - 3)$
6. **For $x^{2}+2x-8$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to - 8 and add up to 2**
The numbers are 4 and - 2 since $4\times(-2)=-8$ and $4+( - 2)=2$.
     - **Step2: Rewrite the middle term and factor by grouping**
$x^{2}+2x-8=x^{2}+4x-2x-8=x(x + 4)-2(x + 4)=(x + 4)(x - 2)$
7. **For $y^{2}-5y-6$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to - 6 and add up to - 5**
The numbers are - 6 and 1 since $(-6)\times1=-6$ and $-6 + 1=-5$.
     - **Step2: Rewrite the middle term and factor by grouping**
$y^{2}-5y-6=y^{2}-6y + y-6=y(y - 6)+1(y - 6)=(y - 6)(y+1)$
8. **For $g^{2}+3g-10$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to - 10 and add up to 3**
The numbers are 5 and - 2 since $5\times(-2)=-10$ and $5+( - 2)=3$.
     - **Step2: Rewrite the middle term and factor by grouping**
$g^{2}+3g-10=g^{2}+5g-2g-10=g(g + 5)-2(g + 5)=(g + 5)(g - 2)$
9. **For $r^{2}+4r-12$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to - 12 and add up to 4**
The numbers are 6 and - 2 since $6\times(-2)=-12$ and $6+( - 2)=4$.
     - **Step2: Rewrite the middle term and factor by grouping**
$r^{2}+4r-12=r^{2}+6r-2r-12=r(r + 6)-2(r + 6)=(r + 6)(r - 2)$
10. **For $x^{2}-x-12$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to - 12 and add up to - 1**
The numbers are - 4 and 3 since $(-4)\times3=-12$ and $-4 + 3=-1$.
     - **Step2: Rewrite the middle term and factor by grouping**
$x^{2}-x-12=x^{2}-4x+3x-12=x(x - 4)+3(x - 4)=(x - 4)(x + 3)$
11. **For $w^{2}-w-6$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to - 6 and add up to - 1**
The numbers are - 3 and 2 since $(-3)\times2=-6$ and $-3 + 2=-1$.
     - **Step2: Rewrite the middle term and factor by grouping**
$w^{2}-w-6=w^{2}-3w+2w-6=w(w - 3)+2(w - 3)=(w - 3)(w + 2)$
12. **For $y^{2}-6y + 8$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to 8 and add up to - 6**
The numbers are - 4 and - 2 since $(-4)\times(-2)=8$ and $-4+( - 2)=-6$.
     - **Step2: Rewrite the middle term and factor by grouping**
$y^{2}-6y + 8=y^{2}-4y-2y + 8=y(y - 4)-2(y - 4)=(y - 4)(y - 2)$
13. **For $x^{2}-8x + 15$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to 15 and add up to - 8**
The numbers are - 5 and - 3 since $(-5)\times(-3)=15$ and $-5+( - 3)=-8$.
     - **Step2: Rewrite the middle term and factor by grouping**
$x^{2}-8x + 15=x^{2}-5x-3x + 15=x(x - 5)-3(x - 5)=(x - 5)(x - 3)$
14. **For $b^{2}-9b + 8$**:
   - **Explanation**:
     - **Step1: Find two - numbers that multiply to 8 and add up to - 9**
The numbers are - 8 and - 1 since $(-8)\times(-1)=8$ and $-8+( - 1)=-9$.
     - **Step2: Rewrite the middle term and factor by grouping**
$b^{2}-9b + 8=b^{2}-8b-b + 8=b(b - 8)-(b - 8)=(b - 8)(b - 1)$

# Answer:
1. $(t + 2)(t + 6)$
2. $(n + 3)(n + 4)$
3. $(p + 4)(p + 5)$
4. $(h + 3)(h + 6)$
5. $(n + 6)(n - 3)$
6. $(x + 4)(x - 2)$
7. $(y - 6)(y + 1)$
8. $(g + 5)(g - 2)$
9. $(r + 6)(r - 2)$
10. $(x - 4)(x + 3)$
11. $(w - 3)(w + 2)$
12. $(y - 4)(y - 2)$
13. $(x - 5)(x - 3)$
14. $(b - 8)(b - 1)$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Answer: