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 - 5.3 Factoring Trinomials of the Type ax2+bx+c - 5.3 Exercise Set: 67

Answer

$(a-2b)(2a-b)$

Work Step by Step

Using the factoring of trinomials in the form $ax^2+bx+c,$ the $\text{ expression }$ \begin{array}{l}\require{cancel} 2a^2-5ab+2b^2 \end{array} has $ac= 2(2)=4 $ and $b= -5 .$ The two numbers with a product of $c$ and a sum of $b$ are $\left\{ -4,-1 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} 2a^2-4ab-1ab+2b^2 .\end{array} Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to \begin{array}{l}\require{cancel} (2a^2-4ab)-(ab-2b^2) .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} 2a(a-2b)-b(a-2b) .\end{array} Factoring the $GCF= (a-2b) $ of the entire expression above results to \begin{array}{l}\require{cancel} (a-2b)(2a-b) .\end{array}
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