Chapter 6.7, Problem 6WE

To determine

To divide: the expression $\frac{a^2-3a+7}{a-2}$.

Answer to Problem 6WE

The quotient is $a-1$ and the remainder is 5.

Explanation of Solution

Given information:

  $\frac{a^2-3a+7}{a-2}$

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 ,

  $\frac{a^2-3a+7}{a-2}$

where

  $\begin{array}{l}\text{divisor}=a-2\\ \text{dividend}=a^2-3a+7\end{array}$

Now, using polynomial division

Step 1.

  $a-2\begin{array}{l}\begin{array}{c}\hfill a\\ \hfill \overline{)\begin{array}{l}{a}^2-3a+7\\ a^2-2a\end{array}}\end{array}\hfill \end{array}$  $\leftarrow \text{Subtract}$

  $-a+7$

Step 2.

  $a-2\begin{array}{l}\begin{array}{c}\hfill a-1\\ \hfill \overline{)\begin{array}{l}-a+7\\ -a+2\end{array}}\end{array}\hfill \end{array}$  $\leftarrow \text{Subtract}$

  $5$

  $\frac{a^2-3a+7}{a-2}=(a-1)+\frac{5}{a-2}$

Thus, the quotient is $a-1$ and the remainder is 5.