From health tips to tech hacks, find it all on IDNLearn.com. Ask anything and receive immediate, well-informed answers from our dedicated community of experts.
Sagot :
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]
### 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]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.