Problem 11 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$y^{2}-8 y+15$$

Step-by-Step Solution

Answer

The factorization of the trinomial \(y^{2}-8 y+15\) is \((y-5)(y-3)\).

Step 1: Identify the Trinomial

The given trinomial is \(y^{2}-8 y+15\). We need to find two numbers that add to -8 (second term) and multiply to 15 (third term).

Step 2: Factorize the Trinomial

Using the concept of number pairs, we find that the numbers -5 and -3 add up to -8 and multiply to 15. Therefore, the trinomial can be factored into \((y-5)(y-3)\).

Step 3: Check the Factorization

To verify, we apply the FOIL method to this product: First: \(y*y =y^{2}\), Outer: \(-5*y = -5y\), Inner: \(-3*y = -3y\), Last: \(-5*-3 = 15\). Adding these four terms up gives \(y^{2} -5y - 3y + 15 = y^{2} -8 y+15\), which matches our initial trinomial, confirming that the factorization is correct.