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Chapter 0: Problem 142

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

Short Answer

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

Step by step solution

01

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.
02

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).
03

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.
04

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.

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