Question

question
factor completely over the integers.
$x^4 + 8x^2 + 7$

Explanation:

Step1: Let \( y = x^2 \)

Substitute \( y \) into the expression, we get \( y^2 + 8y + 7 \).

Step2: Factor the quadratic in \( y \)

We need two numbers that multiply to \( 7 \) and add to \( 8 \). The numbers are \( 1 \) and \( 7 \). So, \( y^2 + 8y + 7=(y + 1)(y + 7) \).

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

Replace \( y \) with \( x^2 \), we have \( (x^2 + 1)(x^2 + 7) \). Since we are factoring over the integers, and \( x^2 + 1 \) and \( x^2 + 7 \) can't be factored further over the integers, this is the complete factorization.

Answer:

\((x^2 + 1)(x^2 + 7)\)