Problem 17 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}-2 x-15$$

Step-by-Step Solution

Answer

The factorization of the trinomial \( x^{2}-2 x-15 \) is \( (x-5)(x+3) \)

Step 1: Identify the coefficients

The coefficients in the trinomial are: a=1 (coefficient of \(x^{2}\)), b=-2 (coefficient of x), and c=-15 (constant term).

Step 2: Find the pair of numbers

The two numbers that add up to -2 (the coefficient of x) and multiply to -15 (the constant term) are -5 and +3. Because (-5)+(+3) = -2 and (-5)*(+3) = -15.

Step 3: Write the factored form

The trinomial factored form is \( (x-5)(x+3) \). This is obtained by splitting the middle term of the trinomial with the pair of numbers found in step 2.

Step 4: Verify using FOIL

FOIL stands for First, Outer, Inner and Last. Applying this to the factored terms we get: First terms: x*x = \( x^{2} \), Outer terms: x*3 = 3x, Inner terms: -5*x = -5x, Last terms: -5*3 = -15. Summing these together we get \( x^{2}+3x-5x-15 = x^{2}-2x-15 \), which matches the original trinomial, thus confirming the factorization is correct.