Chapter 5.2, Problem 48WE

To determine

To find: Blank space in the given statement.

Answer to Problem 48WE

The required polynomial that makes the given statement true is $2+x-3x^2$ .

Explanation of Solution

Given information:

  $2x^{-2}+x^{-1}-3=x^{-2}\left(\underset{\_}{?}\right)$

  $2x^{-2}+x^{-1}-3=x^{-2}\left(\underset{\_}{?}\right)$

  $\begin{array}{l}\Rightarrow 2x^{-2}+x^{-1}-3=2\cdot \frac{1}{x^2}+\frac{1}{x}-3\\ \Rightarrow 2x^{-2}+x^{-1}-3=2\cdot \frac{1}{x^2}+\frac{1}{x}\cdot \frac{x}{x}-3\cdot \frac{x^2}{x^2}\\ \Rightarrow 2x^{-2}+x^{-1}-3=2\cdot \frac{1}{x^2}+\frac{x}{x^{1+1}}-3\cdot \frac{x^2}{x^2}\\ \Rightarrow 2x^{-2}+x^{-1}-3=2\cdot \frac{1}{x^2}+\frac{x}{x^2}-3\cdot \frac{x^2}{x^2}\\ \Rightarrow 2x^{-2}+x^{-1}-3=2\cdot x^{-2}+x\cdot x^{-2}-3\cdot x^2\cdot x^{-2}\\ \Rightarrow 2x^{-2}+x^{-1}-3=x^{-2}\left(2+x-3x^2\right)\end{array}$

Therefore,

The required polynomial that makes the given statement true is $2+x-3x^2$ .