Problem 142 - College Algebra Essentials

Question

Find all integers \(b\) so that the trinomial can be factored. $$x^{2}+b x+15$$

Step-by-Step Solution

Answer

The trinomial \(x^2 + bx + 15\) can be factored for integer values of \(b\) = -8, 8.

Step 1: Understand the condition for factoring the trinomial

A trinomial \(x^2 + bx + c\) can be factored if there exist two integers whose sum equals to \(b\) and product equals to \(c\). In this case, \(c = 15 = 3 * 5\). We're looking for two numbers whose product equals 15.

Step 2: Find the pair(s) of integers

We can find the pairs of integers whose product equals to 15 by checking the pairs of numbers which multiply to 15. These pairs are: (-1, -15), (1, 15), (-3, -5), (3, 5).

Step 3: Check which pairs also sum to \(b\)

Now, we need to find which of these pairs also sum to an integer value. To do this, we can quickly check each pair. Checking each pair, we find that (-3, -5) sums to -8 and (3, 5) sums to 8.

Step 4: Formulate the answer

The pairs that satisfy both conditions are (-3, -5) and (3, 5). Therefore, the trinomial can be factored for \(b\) values of -8 or 8.