Chapter 2.6, Problem 10QR

Solution

To Calculate:

The quotient and remainder when $f\left(x\right)$ is divided by $d\left(x\right)$ .

The quotient $\frac{5}{2}$ and the remainder is $-1$ when $f\left(x\right)$ is divided by $d\left(x\right)$ .

Given:

  $f\left(x\right)=5x-1$

  $d\left(x\right)=2x$

Concepts Used:

If fraction of polynomials $\frac{f\left(x\right)}{d\left(x\right)}$ is expressed as $q\left(x\right)+\frac{r\left(x\right)}{d\left(x\right)}$ , where $q\left(x\right)$ and $r\left(x\right)$ are both polynomials, then $q\left(x\right)$ represents the quotient of division of $f\left(x\right)$ by $d(x)$ and $r\left(x\right)$ represents the remainder.

Calculations:

Express $\frac{f\left(x\right)}{d\left(x\right)}$ as $q\left(x\right)+\frac{r\left(x\right)}{d\left(x\right)}$ and identify the quotient $q\left(x\right)$ and remainder $r\left(x\right)$ .

  $\begin{array}{c}\frac{f\left(x\right)}{d\left(x\right)}=\frac{5x-1}{2x}\\ =\frac{5x}{2x}-\frac{1}{2x}\\ =\frac{5}{2}+\frac{\left(-1\right)}{2x}\\ =\frac{5}{2}+\frac{\left(-1\right)}{d\left(x\right)}\end{array}$

Compare $\frac{5}{2}+\frac{\left(-1\right)}{d\left(x\right)}$ with $q\left(x\right)+\frac{r\left(x\right)}{d\left(x\right)}$ to obtain the quotient $q\left(x\right)=\frac{5}{2}$ and remainder $r\left(x\right)=-1$ .

Conclusion:

The quotient $\frac{5}{2}$ and the remainder is $-1$ when $f\left(x\right)$ is divided by $d\left(x\right)$ .