Question

factor ( x^4 + 11x^2 + 28 ) completely. all factors in your answer should have integer coefficients.

Explanation:

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

The given polynomial is \( x^4 + 11x^2 + 28 \). Let \( y = x^2 \), then the polynomial becomes \( y^2 + 11y + 28 \).

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

We need to find two numbers that multiply to \( 28 \) and add up to \( 11 \). The numbers are \( 4 \) and \( 7 \) because \( 4\times7 = 28 \) and \( 4 + 7 = 11 \). So, \( y^2 + 11y + 28=(y + 4)(y + 7) \).

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

Substituting \( y = x^2 \) into \( (y + 4)(y + 7) \), we get \( (x^2 + 4)(x^2 + 7) \).

Answer:

\( (x^2 + 4)(x^2 + 7) \)