Problem 18 - College Algebra Essentials

Question

Factor each trinomial, or state that the trinomial is prime. $$x^{2}+8 x+15$$

Step-by-Step Solution

Answer

\(x^{2}+8 x+15\) is factorable and it can be rewritten as \((x + 5)(x + 3)\).

Step 1: Identify the structure of the trinomial

We notice that the trinomial \(x^{2}+8 x+15\) is in the form \(ax^2 + bx + c\). Here, a = 1, b = 8 and c = 15.

Step 2: Search for numbers that multiply to 'c' and add to 'b'

We are searching for two numbers that multiply to give c = 15 and add up to give b = 8. By observation, these numbers are 5 and 3 as \(5*3 = 15\) and \(5 + 3 = 8\).

Step 3: Factorize the trinomial

Now, using these two numbers, we can factorize and write the trinomial as two binomials: \(x^{2}+8 x+15 = (x + 5)(x + 3)\).