Certainly! Let's determine which of the given polynomials have a factor of [tex]\(x + 12\)[/tex] by factoring each polynomial step-by-step.
### Given Polynomials:
1. [tex]\(x^2 + 8x + 12\)[/tex]
2. [tex]\(x^2 - 12x + 27\)[/tex]
3. [tex]\(x^2 - 8x - 48\)[/tex]
4. [tex]\(x^2 + 10x - 24\)[/tex]
5. [tex]\(x^2 + 15x + 36\)[/tex]
#### Factoring Each Polynomial:
1. [tex]\(x^2 + 8x + 12\)[/tex]:
[tex]\[
x^2 + 8x + 12 = (x + 2)(x + 6)
\][/tex]
This polynomial factors to [tex]\((x + 2)(x + 6)\)[/tex]. Neither factor is [tex]\(x + 12\)[/tex].
2. [tex]\(x^2 - 12x + 27\)[/tex]:
[tex]\[
x^2 - 12x + 27 = (x - 3)(x - 9)
\][/tex]
This polynomial factors to [tex]\((x - 3)(x - 9)\)[/tex]. Neither factor is [tex]\(x + 12\)[/tex].
3. [tex]\(x^2 - 8x - 48\)[/tex]:
[tex]\[
x^2 - 8x - 48 = (x - 12)(x + 4)
\][/tex]
This polynomial factors to [tex]\((x - 12)(x + 4)\)[/tex]. Neither factor is [tex]\(x + 12\)[/tex].
4. [tex]\(x^2 + 10x - 24\)[/tex]:
[tex]\[
x^2 + 10x - 24 = (x + 12)(x - 2)
\][/tex]
This polynomial factors to [tex]\((x + 12)(x - 2)\)[/tex]. One of the factors is [tex]\(x + 12\)[/tex].
5. [tex]\(x^2 + 15x + 36\)[/tex]:
[tex]\[
x^2 + 15x + 36 = (x + 3)(x + 12)
\][/tex]
This polynomial factors to [tex]\((x + 3)(x + 12)\)[/tex]. One of the factors is [tex]\(x + 12\)[/tex].
### Conclusion
Among the given polynomials, the ones that have a factor of [tex]\(x + 12\)[/tex] are:
- [tex]\(x^2 + 10x - 24\)[/tex], which factors to [tex]\((x + 12)(x - 2)\)[/tex]
- [tex]\(x^2 + 15x + 36\)[/tex], which factors to [tex]\((x + 3)(x + 12)\)[/tex]
Thus, the polynomials that have a factor of [tex]\(x + 12\)[/tex] are:
[tex]\[ x^2 + 10x - 24 \][/tex]
[tex]\[ x^2 + 15x + 36 \][/tex]
