Chapter 2.6, Problem 9QR

Solution

To Calculate:

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

The quotient $3$ and the remainder is $-5$ when $f\left(x\right)$ is divided by $d\left(x\right)$ .

Given:

  $f\left(x\right)=3x-5$

  $d\left(x\right)=x$

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 $r\left(x\right)$ .

  $\begin{array}{c}\frac{f\left(x\right)}{d\left(x\right)}=\frac{3x-5}{x}\\ =\frac{3x}{x}-\frac{5}{x}\\ =3+\frac{\left(-5\right)}{x}\\ =3+\frac{\left(-5\right)}{d\left(x\right)}\end{array}$

Compare $3+\frac{\left(-5\right)}{d\left(x\right)}$ with $q\left(x\right)+\frac{r\left(x\right)}{d\left(x\right)}$ to obtain the quotient $q\left(x\right)=3$ and remainder $r\left(x\right)=-5$ .

Conclusion:

The quotient $3$ and the remainder is $-5$ when $f\left(x\right)$ is divided by $d\left(x\right)$ .