Question

factor the trinomial completely.

(a^4 - 5a^2 - 14)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. (a^4 - 5a^2 - 14 = ) (factor completely.)
b. the polynomial is prime.

Explanation:

🆕 New Concept Discovered: Factoring Trinomials in Quadratic Form
Treating higher-degree expressions like standard quadratics

Step 1: Substitute to find quadratic form

Let \( u = a^2 \). Then \( u^2 = a^4 \).

Rewrite the expression in terms of \( u \):

$$ u^2 - 5u - 14 $$

Step 2: Factor the quadratic expression

Find two numbers that multiply to \(-14\) and add to \(-5\). These numbers are \(-7\) and \(2\).

$$ (u - 7)(u + 2) $$

Step 3: Substitute back

Replace \( u \) with \( a^2 \):

$$ (a^2 - 7)(a^2 + 2) $$

Neither binomial can be factored further over the integers.

Answer:

A. \( a^4 - 5a^2 - 14 = (a^2 - 7)(a^2 + 2) \)