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To determine which polynomials have a factor of [tex]\( x + 12 \)[/tex], we need to factor each polynomial and see if [tex]\( x + 12 \)[/tex] is one of the factors.
Let's factor each polynomial step by step.
### Polynomial 1: [tex]\( x^2 + 8x + 12 \)[/tex]
To factor [tex]\( x^2 + 8x + 12 \)[/tex]:
1. Find two numbers that multiply to 12 (the constant term) and add up to 8 (the coefficient of the [tex]\( x \)[/tex] term).
Those numbers are 2 and 6 because [tex]\( 2 \times 6 = 12 \)[/tex] and [tex]\( 2 + 6 = 8 \)[/tex].
2. Write the polynomial as:
[tex]\[ (x + 2)(x + 6) \][/tex]
So, [tex]\( x^2 + 8x + 12 = (x + 2)(x + 6) \)[/tex].
### Polynomial 2: [tex]\( x^2 + 15x + 36 \)[/tex]
To factor [tex]\( x^2 + 15x + 36 \)[/tex]:
1. Find two numbers that multiply to 36 and add up to 15.
Those numbers are 3 and 12 because [tex]\( 3 \times 12 = 36 \)[/tex] and [tex]\( 3 + 12 = 15 \)[/tex].
2. Write the polynomial as:
[tex]\[ (x + 3)(x + 12) \][/tex]
So, [tex]\( x^2 + 15x + 36 = (x + 3)(x + 12) \)[/tex].
### Polynomial 3: [tex]\( x^2 - 12x + 27 \)[/tex]
To factor [tex]\( x^2 - 12x + 27 \)[/tex]:
1. Find two numbers that multiply to 27 and add up to -12.
Those numbers are -3 and -9 because [tex]\( -3 \times -9 = 27 \)[/tex] and [tex]\( -3 + (-9) = -12 \)[/tex].
2. Write the polynomial as:
[tex]\[ (x - 3)(x - 9) \][/tex]
So, [tex]\( x^2 - 12x + 27 = (x - 3)(x - 9) \)[/tex].
### Polynomial 4: [tex]\( x^2 + 10x - 24 \)[/tex]
To factor [tex]\( x^2 + 10x - 24 \)[/tex]:
1. Find two numbers that multiply to -24 and add up to 10.
Those numbers are 12 and -2 because [tex]\( 12 \times -2 = -24 \)[/tex] and [tex]\( 12 + (-2) = 10 \)[/tex].
2. Write the polynomial as:
[tex]\[ (x + 12)(x - 2) \][/tex]
So, [tex]\( x^2 + 10x - 24 = (x + 12)(x - 2) \)[/tex].
### Polynomial 5: [tex]\( x^2 - 8x - 48 \)[/tex]
To factor [tex]\( x^2 - 8x - 48 \)[/tex]:
1. Find two numbers that multiply to -48 and add up to -8.
Those numbers are -12 and 4 because [tex]\( -12 \times 4 = -48 \)[/tex] and [tex]\( -12 + 4 = -8 \)[/tex].
2. Write the polynomial as:
[tex]\[ (x - 12)(x + 4) \][/tex]
So, [tex]\( x^2 - 8x - 48 = (x - 12)(x + 4) \)[/tex].
### Conclusion:
The polynomials that have a factor of [tex]\( x + 12 \)[/tex] are:
- [tex]\( x^2 + 15x + 36 = (x + 3)(x + 12) \)[/tex]
- [tex]\( x^2 + 10x - 24 = (x + 12)(x - 2) \)[/tex]
Thus, the two polynomials with a factor of [tex]\( x + 12 \)[/tex] are:
1. [tex]\( x^2 + 15x + 36 \)[/tex]
2. [tex]\( x^2 + 10x - 24 \)[/tex]
Let's factor each polynomial step by step.
### Polynomial 1: [tex]\( x^2 + 8x + 12 \)[/tex]
To factor [tex]\( x^2 + 8x + 12 \)[/tex]:
1. Find two numbers that multiply to 12 (the constant term) and add up to 8 (the coefficient of the [tex]\( x \)[/tex] term).
Those numbers are 2 and 6 because [tex]\( 2 \times 6 = 12 \)[/tex] and [tex]\( 2 + 6 = 8 \)[/tex].
2. Write the polynomial as:
[tex]\[ (x + 2)(x + 6) \][/tex]
So, [tex]\( x^2 + 8x + 12 = (x + 2)(x + 6) \)[/tex].
### Polynomial 2: [tex]\( x^2 + 15x + 36 \)[/tex]
To factor [tex]\( x^2 + 15x + 36 \)[/tex]:
1. Find two numbers that multiply to 36 and add up to 15.
Those numbers are 3 and 12 because [tex]\( 3 \times 12 = 36 \)[/tex] and [tex]\( 3 + 12 = 15 \)[/tex].
2. Write the polynomial as:
[tex]\[ (x + 3)(x + 12) \][/tex]
So, [tex]\( x^2 + 15x + 36 = (x + 3)(x + 12) \)[/tex].
### Polynomial 3: [tex]\( x^2 - 12x + 27 \)[/tex]
To factor [tex]\( x^2 - 12x + 27 \)[/tex]:
1. Find two numbers that multiply to 27 and add up to -12.
Those numbers are -3 and -9 because [tex]\( -3 \times -9 = 27 \)[/tex] and [tex]\( -3 + (-9) = -12 \)[/tex].
2. Write the polynomial as:
[tex]\[ (x - 3)(x - 9) \][/tex]
So, [tex]\( x^2 - 12x + 27 = (x - 3)(x - 9) \)[/tex].
### Polynomial 4: [tex]\( x^2 + 10x - 24 \)[/tex]
To factor [tex]\( x^2 + 10x - 24 \)[/tex]:
1. Find two numbers that multiply to -24 and add up to 10.
Those numbers are 12 and -2 because [tex]\( 12 \times -2 = -24 \)[/tex] and [tex]\( 12 + (-2) = 10 \)[/tex].
2. Write the polynomial as:
[tex]\[ (x + 12)(x - 2) \][/tex]
So, [tex]\( x^2 + 10x - 24 = (x + 12)(x - 2) \)[/tex].
### Polynomial 5: [tex]\( x^2 - 8x - 48 \)[/tex]
To factor [tex]\( x^2 - 8x - 48 \)[/tex]:
1. Find two numbers that multiply to -48 and add up to -8.
Those numbers are -12 and 4 because [tex]\( -12 \times 4 = -48 \)[/tex] and [tex]\( -12 + 4 = -8 \)[/tex].
2. Write the polynomial as:
[tex]\[ (x - 12)(x + 4) \][/tex]
So, [tex]\( x^2 - 8x - 48 = (x - 12)(x + 4) \)[/tex].
### Conclusion:
The polynomials that have a factor of [tex]\( x + 12 \)[/tex] are:
- [tex]\( x^2 + 15x + 36 = (x + 3)(x + 12) \)[/tex]
- [tex]\( x^2 + 10x - 24 = (x + 12)(x - 2) \)[/tex]
Thus, the two polynomials with a factor of [tex]\( x + 12 \)[/tex] are:
1. [tex]\( x^2 + 15x + 36 \)[/tex]
2. [tex]\( x^2 + 10x - 24 \)[/tex]
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