Problem 87 - Introductory Algebra for College Students

Question

In factoring \(x^{2}+b x+c,\) describe how the last terms in each factor are related to \(b\) and \(c .\)

Step-by-Step Solution

Answer

The last terms ( \(p\) and \(q\) ) in each factor of quadratic expression \(x^2 + bx + c\) relate to \(b\) and \(c\) respectively such that \(b = p + q\) (sum of \(p\) and \(q\)) and \(c = pq\) (product of \(p\) and \(q\)).

Step 1: Recognize the Quadratic Form

The first step is recognizing the expression as a general form of a quadratic equation, \(x^2 + bx + c\). The values of \(b\) and \(c\) can impact the factors in a crucial manner.

Step 2: Put in Factored Form

Assume that we factorize the quadratic equation into form, \(x^2 + bx + c = (x + p)(x + q)\). Here, \(p\) and \(q\) are the last terms in each factor.

Step 3: Find Expression for \(p\) and \(q\)

By multiplying out these factors, we get \(x^2 + px + qx + pq\), which needs to be equivalent to \(x^2 + bx + c\). Comparing the like terms, we see \(b = p + q\) (the sum of \(p\) and \(q\)), and \(c = pq\) (the product of \(p\) and \(q\)).