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: 51

Answer

$\left( t+\dfrac{1}{3}\right)^2$

Work Step by Step

The 2 numbers whose product is $ac= 1\left( \dfrac{1}{9} \right)=\dfrac{1}{9} $ and whose sum is $b= \dfrac{2}{3} $ are $\left\{ \dfrac{1}{3}, \dfrac{1}{3} \right\}.$ Using these two numbers to decompose the middle term of the given trinomial, $ t^2+\dfrac{2}{3}t+\dfrac{1}{9} ,$ then the factored form is \begin{array}{l} t^2+\dfrac{1}{3}t+\dfrac{1}{3}t+\dfrac{1}{9} \\\\= \left( t^2+\dfrac{1}{3}t \right) +\left( \dfrac{1}{3}t+\dfrac{1}{9}\right) \\\\= t\left( t+\dfrac{1}{3} \right) +\dfrac{1}{3}\left( t+\dfrac{1}{3}\right) \\\\= \left( t+\dfrac{1}{3}\right)\left( t+\dfrac{1}{3}\right) \\\\= \left( t+\dfrac{1}{3}\right)^2 .\end{array}
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