Problem 14 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}+3 x-28$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^{2}+3x-28\) is \((x+7)(x-4)\).

Step 1: Define the Trinomial

The trinomial we are working on is \(x^{2}+3x-28\).

Step 2: Factoring the Trinomial

We need to find two numbers that multiply to -28 (the constant term), and add to 3 (the coefficient of the x term). After consideration, 7 and -4 fit these criteria. Thus, the factored form of the trinomial is \((x+7)(x-4)\).

Step 3: Checking the Factorization Using FOIL

Let us use the FOIL method to expand the factored form and confirm that we obtain the original trinomial: \nFirst terms: \(x*x = x^{2}\), \nOuter terms: \(x*(-4) = -4x\), \nInner terms: \(7*x= 7x\), \nLast terms: \(7*(-4) = -28\).\nAdding these together we obtain \(x^{2}+3x-28\), which matches the given trinomial.