Chapter 6.7, Problem 13WE

To determine

To divide: the expression $\frac{3x^2+10x-9}{3x-2}$.

Answer to Problem 13WE

The quotient is $x+4$ and the remainder is $(-1)$.

Explanation of Solution

Given information:

  $\frac{3x^2+10x-9}{3x-2}$

Formula used:

  $\frac{\text{Dividend}}{\text{Divisor}}\text{=Quotient+}\frac{\text{Remainder}}{\text{Divisor}}$ .

Or

  $\text{Dividend=Quotient×Divisor+Remainder}$

Dividing the polynomials , each polynomial to be arranged in decreasing degree of the variable.

The missing terms in decreasing degree of variable to filled by using zero coefficients.

Calculation:

Consider ,

  $\frac{3x^2+10x-9}{3x-2}$

where

  $\begin{array}{l}\text{divisor}=3x-2\\ \text{dividend}=3x^2+10x-9\end{array}$

Now, using polynomial division

Step 1.

  $3x-2\begin{array}{l}\begin{array}{c}\hfill x\\ \hfill \overline{)\begin{array}{l}3x^2+10x-9\\ 3x^2-2x\end{array}}\end{array}\hfill \end{array}$  $\leftarrow \text{Subtract}$

  $12x-9$

Step 2.

  $3x-2\begin{array}{l}\begin{array}{c}\hfill x+4\\ \hfill \overline{)\begin{array}{l}12x-9\\ 12x-8\end{array}}\end{array}\hfill \end{array}$  $\leftarrow \text{Subtract}$

  $-1$

  $\frac{3x^2+10x-9}{3x-2}=(x+4)+\frac{(-1)}{3x-2}$

Thus, the quotient is $x+4$ and the remainder is $(-1)$.