To find the completely factored form of the polynomial [tex]\( x^4 + 12x^2 + 32 \)[/tex], we can follow a systematic approach to factor it step-by-step.
Step 1: Identify the polynomial.
We start with the polynomial:
[tex]\[ x^4 + 12x^2 + 32 \][/tex]
Step 2: Look for patterns.
Notice that the polynomial can be viewed as a quadratic in terms of [tex]\( x^2 \)[/tex]. Let [tex]\( y = x^2 \)[/tex], then the polynomial becomes:
[tex]\[ y^2 + 12y + 32 \][/tex]
Step 3: Factor the quadratic expression.
Now we need to factor the quadratic in terms of [tex]\( y \)[/tex]:
[tex]\[ y^2 + 12y + 32 \][/tex]
We look for two numbers that multiply to 32 and add up to 12. These numbers are 4 and 8.
So, we can factor the quadratic as:
[tex]\[ y^2 + 12y + 32 = (y + 4)(y + 8) \][/tex]
Step 4: Substitute [tex]\( y \)[/tex] back with [tex]\( x^2 \)[/tex].
Remembering that [tex]\( y = x^2 \)[/tex], we substitute back:
[tex]\[ (x^2 + 4)(x^2 + 8) \][/tex]
Step 5: Verify our factorization.
[tex]\[
(x^2 + 4)(x^2 + 8)
\][/tex]
Expanding this product should give us back the original polynomial:
[tex]\[
(x^2 + 4)(x^2 + 8) = x^4 + 8x^2 + 4x^2 + 32 = x^4 + 12x^2 + 32
\][/tex]
So, our factorization is verified.
Therefore, the completely factored form of the polynomial [tex]\( x^4 + 12x^2 + 32 \)[/tex] is:
[tex]\[ \left(x^2 + 4\right)\left(x^2 + 8\right) \][/tex]
The correct answer is:
B. [tex]\(\left(x^2 + 4\right)\left(x^2 + 8\right)\)[/tex]
