Question

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

Explanation:

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

The given polynomial is \( x^4 + 12x^2 + 32 \). Let \( y=x^2 \), then the polynomial becomes \( y^2 + 12y + 32 \).

Step2: Factor the quadratic in \( y \)

We need to factor \( y^2 + 12y + 32 \). We look for two numbers that multiply to \( 32 \) and add up to \( 12 \). The numbers are \( 4 \) and \( 8 \), so \( y^2 + 12y + 32=(y + 4)(y + 8) \).

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

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

Answer:

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