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When written in factored form, which two polynomials have a factor of [tex]x + 12[/tex]?

A. [tex]x^2 + 15x + 36[/tex]
B. [tex]x^2 - 8x - 48[/tex]
C. [tex]x^2 + 8x + 12[/tex]
D. [tex]x^2 - 12x + 27[/tex]
E. [tex]x^2 + 10x - 24[/tex]


Sagot :

To determine which polynomials have a factor of [tex]\( x+12 \)[/tex], we will factor each polynomial individually and look for [tex]\( x+12 \)[/tex] among the factors.

### 1. [tex]\( x^2 + 15x + 36 \)[/tex]

To factor [tex]\( x^2 + 15x + 36 \)[/tex], we need to find two numbers that multiply to 36 and add to 15. Those numbers are 3 and 12.

[tex]\[ x^2 + 15x + 36 = (x+3)(x+12) \][/tex]

The factor is [tex]\( x + 12 \)[/tex].

### 2. [tex]\( x^2 - 8x - 48 \)[/tex]

To factor [tex]\( x^2 - 8x - 48 \)[/tex], we need to find two numbers that multiply to -48 and add to -8. Those numbers are -12 and 4.

[tex]\[ x^2 - 8x - 48 = (x-12)(x+4) \][/tex]

The factor is not [tex]\( x + 12 \)[/tex].

### 3. [tex]\( x^2 + 8x + 12 \)[/tex]

To factor [tex]\( x^2 + 8x + 12 \)[/tex], we need to find two numbers that multiply to 12 and add to 8. Those numbers are 2 and 6.

[tex]\[ x^2 + 8x + 12 = (x+2)(x+6) \][/tex]

The factor is not [tex]\( x + 12 \)[/tex].

### 4. [tex]\( x^2 - 12x + 27 \)[/tex]

To factor [tex]\( x^2 - 12x + 27 \)[/tex], we need to find two numbers that multiply to 27 and add to -12. Those numbers are -3 and -9.

[tex]\[ x^2 - 12x + 27 = (x-3)(x-9) \][/tex]

The factor is not [tex]\( x + 12 \)[/tex].

### 5. [tex]\( x^2 + 10x - 24 \)[/tex]

To factor [tex]\( x^2 + 10x - 24 \)[/tex], we need to find two numbers that multiply to -24 and add to 10. Those numbers are 12 and -2.

[tex]\[ x^2 + 10x - 24 = (x+12)(x-2) \][/tex]

The factor is [tex]\( x + 12 \)[/tex].

### Conclusion

The polynomials [tex]\( x^2 + 15x + 36 \)[/tex] and [tex]\( x^2 + 10x - 24 \)[/tex] have a factor of [tex]\( x + 12 \)[/tex].