Question

factor ( x^4 + 15x^2 + 50 ) completely. all factors in your answer should have integer coefficients.

Explanation:

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

The given polynomial is \( x^4 + 15x^2 + 50 \). Let \( y = x^2 \), then the polynomial becomes \( y^2 + 15y + 50 \).

Step2: Factor the quadratic in \( y \)

We need to find two numbers that multiply to \( 50 \) and add up to \( 15 \). The numbers \( 5 \) and \( 10 \) work because \( 5\times10 = 50 \) and \( 5 + 10 = 15 \). So, \( y^2 + 15y + 50=(y + 5)(y + 10) \).

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

Substituting \( y = x^2 \) back into the factored form, we get \( (x^2 + 5)(x^2 + 10) \). Since \( x^2 + 5 \) and \( x^2 + 10 \) cannot be factored further over the integers (their discriminants \( -20 \) and \( -40 \) are negative and not perfect squares), this is the complete factorization.

Answer:

\((x^2 + 5)(x^2 + 10)\)