Question

select the correct answer
what is the completely factored form of this polynomial?
x⁴ + 12x² + 32
a. (x² + 4)(x + 2)(x + 4)
b. (x² + 4)(x² + 8)
c. (x + 4)(x + 8)
d. (x + 2)(x - 2)(x² + 8)

Explanation:

Step1: Let \( y = x^2 \), rewrite the polynomial.

The polynomial \( x^4 + 12x^2 + 32 \) can be rewritten as \( y^2 + 12y + 32 \) by substituting \( y = x^2 \).

Step2: Factor the quadratic in \( y \).

We need to find two numbers that multiply to \( 32 \) and add up to \( 12 \). The numbers are \( 4 \) and \( 8 \) because \( 4\times8 = 32 \) and \( 4 + 8 = 12 \). So, \( y^2 + 12y + 32=(y + 4)(y + 8) \).

Step3: Substitute back \( y = x^2 \).

Substituting \( y = x^2 \) back into the factored form, we get \( (x^2 + 4)(x^2 + 8) \).

Answer:

B. \((x^{2}+4)(x^{2}+8)\)