Problem 15 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$y^{2}+10 y-39$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(y^{2}+10y-39\) is \((y-3)(y+13)\).

Step 1: Identify the trinomial

The given trinomial is \(y^{2}+10y-39\). The goal is to find factors \(m\) and \(n\), of the term \(-39\) that sum to \(10\).

Step 2: Factor the trinomial

The factors of \(-39\) that add up to \(10\) are \(-3\) and \(13\). Therefore, the trinomial can be factored to \((y-3)(y+13)\).

Step 3: Check the factorization using FOIL method

We validate by using the FOIL method: First terms: \(y*y = y^{2}\), Outer terms: \(y*13 = 13y\), Inner terms: \(-3*y = -3y\), Last terms: \(-3*13 = -39\). Summing these up: \(y^{2} + 13y - 3y - 39 = y^{2} + 10y - 39\), which is indeed the original trinomial. So, the factorization is correct and the check is successful.