Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 5 - Polynomials and Factoring - 5.2 Factoring Trinomials of the Type x2+bx+c - 5.2 Exercise Set: 5

Answer

positive; positive

Work Step by Step

Given: $x^2+bx+c=(x+p)(x+q)$ Note that when multiplied using FOIL method, $(x+p)(x+q) \\= x(x)+xq+px+pq \\=x^2+(p+q)x=pq$ Thus, $x^2+bx+c=(x+p)(x+q) = x^2+(p+q)x+pq$ This means that $b=p+q \\c=pq$ In $c=pq$, if $c$ is positive, then either both $p$ and $q$ are positive or both are negative (since the product of two numbers is positive only when the two numbers have the same sign). However, since $b$ is also positive, then $p$ and $q$ cannot be negative; otherwise, their sum, $p+q$, which is known to be equal to $b$, will also be negative. Therefore, if both $b$ and $c$ are positive, then $p$ and $q$ must also be both positive.
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