Problem 28 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$w^{2}+12 w-64$$

Step-by-Step Solution

Answer

The factorization of the trinomial \(w^{2} + 12w - 64\) is \((w - 4)(w + 16)\).

Step 1: Identify the Form of the Trinomial

The trinomial is in the form \(ax^{2} + bx + c\) where \(a = 1, b = 12, and c = -64\). The goal is to find two numbers that multiply to \(c\) and add up to \(b\).

Step 2: Identify the Factors

The numbers 16 and -4 are suitable because 16 times -4 equals -64, and 16 plus -4 equals 12.

Step 3: Write the Factorization

Therefore, the trinomial \(w^{2} + 12w - 64\) can be factorized as \((w - 4)(w + 16)\).

Step 4: Check by Multiplying

Check this factorization using FOIL multiplication. If the factors are correct, they will multiply back into the original trinomial: \((w - 4)(w + 16) = w^2 + 16w - 4w - 64 = w^2 + 12w - 64\). Thus, the factorization is correct.