Problem 20 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}-5 x-6$$

Step-by-Step Solution

Answer

The trinomial \(x^{2}-5 x-6\) factors to \( (x-6)(x+1) \)

Step 1: Factoring the Trinomial

Look for two numbers that multiply to -6 (the third term in the trinomial) and add up to -5 (the coefficient of the second term). After consideration, these would be -6 and 1, because -6 multiplied by 1 equals -6, and -6 added to 1 equals -5. Therefore, the trinomial can be factored as: \( (x - 6)(x + 1) \)

Step 2: Checking the Factorization with FOIL

Next, confirm the factorization using the FOIL (First, Outside, Inside, Last) method. Multiply \(x-6\) by \(x+1\):\n- First: Multiply the first terms in each binomial: \(x * x = x^{2}\)\n- Outside: Multiply the outside terms in the product: \(x * 1 = x\) \n- Inside: Multiply the inside terms in the product: \(-6 * x = -6x\) \n- Last: Multiply the last terms in each binomial: \(-6 * 1 = -6\)\nCollecting these results gives the original trinomial: \( x^{2} + x - 6x - 6 \) which simplifies to \( x^{2} - 5x - 6 \)