Problem 42 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$a^{2}-18 a b+80 b^{2}$$

Step-by-Step Solution

Answer

The factor form of the trinomial \(a^{2}-18 a b+80 b^{2}\) is \((a -10b)(a - 8b)\).

Step 1: Identify Trinomial Form

The given trinomial \(a^{2}-18 a b+80 b^{2}\) can be written in the general form of a trinomial, \(ax^{2}+bx+c\), where \(a = 1\), \(b = -18\), and \(c = 80\).

Step 2: Factoring The Trinomial

To factor the trinomial, we need to find two numbers that multiply to \(a*c(=1*80=80)\) and add up to \(b(-18)\). Numbers that satisfy these conditions are -10 and -8. Therefore, the factors can be written down as: \((a -10b)(a - 8b)\).

Step 3: Check Factorization by FOIL

Use the FOIL method on the factors identified in Step 2 to verify the factorization. FOIL involves multiplying the terms in the following order: First terms: \(a * a = a^{2}\), Outer terms: \(a * -8b = -8ab\), Inner terms: \(-10b * a = -10ab\), Last terms: \(-10b * -8b = 80 b^{2}\). Summing these up gives \(a^{2}-18 a b+80 b^{2}\), which matches the original trinomial, confirming the correctness of the factorization.