Problem 18 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}-4 x-5$$

Step-by-Step Solution

Answer

The factorization is \((x-5)(x+1)\).

Step 1: Identify numbers

Find two numbers that multiply to -5 and add to -4. These numbers are -5 and 1, because -5 times 1 is -5, and -5 plus 1 is -4.

Step 2: Factor the trinomial

Write the quadratic trinomial as a product of two binomials. Using the numbers from the previous step, we have: \(x^{2}-4 x-5\) equals \((x-5)(x+1)\).

Step 3: Verification using FOIL

Verify the factorization using FOIL (First, Outer, Inner, Last) multiplication. Multiply the first terms: \(x \cdot x = x^{2}\). Multiply the outer terms: \(x \cdot 1 = x\). Multiply the inner terms: \(-5 \cdot x = -5x\). Multiply the last terms: \(-5 \cdot 1 = -5\). Combine like terms: \(x^{2} -4x -5\), which matches with the original trinomial, confirming the factorization.