Answer :
To determine which polynomials have a factor of [tex]\(x+12\)[/tex], let's consider each polynomial and identify the factors. We will factorize each of the given quadratic polynomials one by one.
1. Factorizing [tex]\(x^2 + 15x + 36\)[/tex]:
The roots of the polynomial can be found by solving the equation [tex]\(x^2 + 15x + 36 = 0\)[/tex].
We look for two numbers that multiply to [tex]\(36\)[/tex] and add up to [tex]\(15\)[/tex].
These numbers are [tex]\(3\)[/tex] and [tex]\(12\)[/tex]. Therefore, we can write:
[tex]\[ x^2 + 15x + 36 = (x + 3)(x + 12) \][/tex]
2. Factorizing [tex]\(x^2 - 8x - 48\)[/tex]:
The roots of the polynomial can be found by solving the equation [tex]\(x^2 - 8x - 48 = 0\)[/tex].
We look for two numbers that multiply to [tex]\(-48\)[/tex] and add up to [tex]\(-8\)[/tex].
These numbers are [tex]\(-12\)[/tex] and [tex]\(4\)[/tex]. Therefore, we can write:
[tex]\[ x^2 - 8x - 48 = (x - 12)(x + 4) \][/tex]
3. Factorizing [tex]\(x^2 + 8x + 12\)[/tex]:
The roots of the polynomial can be found by solving the equation [tex]\(x^2 + 8x + 12 = 0\)[/tex].
We look for two numbers that multiply to [tex]\(12\)[/tex] and add up to [tex]\(8\)[/tex].
These numbers are [tex]\(2\)[/tex] and [tex]\(6\)[/tex]. Therefore, we can write:
[tex]\[ x^2 + 8x + 12 = (x + 2)(x + 6) \][/tex]
4. Factorizing [tex]\(x^2 - 12x + 27\)[/tex]:
The roots of the polynomial can be found by solving the equation [tex]\(x^2 - 12x + 27 = 0\)[/tex].
We look for two numbers that multiply to [tex]\(27\)[/tex] and add up to [tex]\(-12\)[/tex].
These numbers are [tex]\(-3\)[/tex] and [tex]\(-9\)[/tex]. Therefore, we can write:
[tex]\[ x^2 - 12x + 27 = (x - 3)(x - 9) \][/tex]
5. Factorizing [tex]\(x^2 + 10x - 24\)[/tex]:
The roots of the polynomial can be found by solving the equation [tex]\(x^2 + 10x - 24 = 0\)[/tex].
We look for two numbers that multiply to [tex]\(-24\)[/tex] and add up to [tex]\(10\)[/tex].
These numbers are [tex]\(12\)[/tex] and [tex]\(-2\)[/tex]. Therefore, we can write:
[tex]\[ x^2 + 10x - 24 = (x + 12)(x - 2) \][/tex]
Next, we identify which of these factorizations include the factor [tex]\(x + 12\)[/tex]:
- From [tex]\(x^2 + 15x + 36\)[/tex] we have [tex]\((x + 3)(x + 12)\)[/tex], which includes [tex]\(x + 12\)[/tex].
- From [tex]\(x^2 - 8x - 48\)[/tex] we have [tex]\((x - 12)(x + 4)\)[/tex], which does not include [tex]\(x + 12\)[/tex].
- From [tex]\(x^2 + 8x + 12\)[/tex] we have [tex]\((x + 2)(x + 6)\)[/tex], which does not include [tex]\(x + 12\)[/tex].
- From [tex]\(x^2 - 12x + 27\)[/tex] we have [tex]\((x - 3)(x - 9)\)[/tex], which does not include [tex]\(x + 12\)[/tex].
- From [tex]\(x^2 + 10x - 24\)[/tex] we have [tex]\((x + 12)(x - 2)\)[/tex], which includes [tex]\(x + 12\)[/tex].
Thus, the polynomials that have [tex]\(x + 12\)[/tex] as a factor are [tex]\(x^2 + 15x + 36\)[/tex] and [tex]\(x^2 + 10x - 24\)[/tex].
1. Factorizing [tex]\(x^2 + 15x + 36\)[/tex]:
The roots of the polynomial can be found by solving the equation [tex]\(x^2 + 15x + 36 = 0\)[/tex].
We look for two numbers that multiply to [tex]\(36\)[/tex] and add up to [tex]\(15\)[/tex].
These numbers are [tex]\(3\)[/tex] and [tex]\(12\)[/tex]. Therefore, we can write:
[tex]\[ x^2 + 15x + 36 = (x + 3)(x + 12) \][/tex]
2. Factorizing [tex]\(x^2 - 8x - 48\)[/tex]:
The roots of the polynomial can be found by solving the equation [tex]\(x^2 - 8x - 48 = 0\)[/tex].
We look for two numbers that multiply to [tex]\(-48\)[/tex] and add up to [tex]\(-8\)[/tex].
These numbers are [tex]\(-12\)[/tex] and [tex]\(4\)[/tex]. Therefore, we can write:
[tex]\[ x^2 - 8x - 48 = (x - 12)(x + 4) \][/tex]
3. Factorizing [tex]\(x^2 + 8x + 12\)[/tex]:
The roots of the polynomial can be found by solving the equation [tex]\(x^2 + 8x + 12 = 0\)[/tex].
We look for two numbers that multiply to [tex]\(12\)[/tex] and add up to [tex]\(8\)[/tex].
These numbers are [tex]\(2\)[/tex] and [tex]\(6\)[/tex]. Therefore, we can write:
[tex]\[ x^2 + 8x + 12 = (x + 2)(x + 6) \][/tex]
4. Factorizing [tex]\(x^2 - 12x + 27\)[/tex]:
The roots of the polynomial can be found by solving the equation [tex]\(x^2 - 12x + 27 = 0\)[/tex].
We look for two numbers that multiply to [tex]\(27\)[/tex] and add up to [tex]\(-12\)[/tex].
These numbers are [tex]\(-3\)[/tex] and [tex]\(-9\)[/tex]. Therefore, we can write:
[tex]\[ x^2 - 12x + 27 = (x - 3)(x - 9) \][/tex]
5. Factorizing [tex]\(x^2 + 10x - 24\)[/tex]:
The roots of the polynomial can be found by solving the equation [tex]\(x^2 + 10x - 24 = 0\)[/tex].
We look for two numbers that multiply to [tex]\(-24\)[/tex] and add up to [tex]\(10\)[/tex].
These numbers are [tex]\(12\)[/tex] and [tex]\(-2\)[/tex]. Therefore, we can write:
[tex]\[ x^2 + 10x - 24 = (x + 12)(x - 2) \][/tex]
Next, we identify which of these factorizations include the factor [tex]\(x + 12\)[/tex]:
- From [tex]\(x^2 + 15x + 36\)[/tex] we have [tex]\((x + 3)(x + 12)\)[/tex], which includes [tex]\(x + 12\)[/tex].
- From [tex]\(x^2 - 8x - 48\)[/tex] we have [tex]\((x - 12)(x + 4)\)[/tex], which does not include [tex]\(x + 12\)[/tex].
- From [tex]\(x^2 + 8x + 12\)[/tex] we have [tex]\((x + 2)(x + 6)\)[/tex], which does not include [tex]\(x + 12\)[/tex].
- From [tex]\(x^2 - 12x + 27\)[/tex] we have [tex]\((x - 3)(x - 9)\)[/tex], which does not include [tex]\(x + 12\)[/tex].
- From [tex]\(x^2 + 10x - 24\)[/tex] we have [tex]\((x + 12)(x - 2)\)[/tex], which includes [tex]\(x + 12\)[/tex].
Thus, the polynomials that have [tex]\(x + 12\)[/tex] as a factor are [tex]\(x^2 + 15x + 36\)[/tex] and [tex]\(x^2 + 10x - 24\)[/tex].
