Chapter A.3, Problem 71AYU

To determine

To find: The quotient and the remainder and verify the answer.

Answer to Problem 71AYU

The quotient is $x^2+ax+a^2$ and remainder is $0$ .

Explanation of Solution

Given information:

  $\text{Divide}\left(x^3-a^3\right)\text{by}\left(x-a\right)$ .

Calculation:

Divide the first term of dividend by the first term of divisor and use it as the first term of quotient.

  $x-a\begin{array}{c}\hfill x^2\\ \hfill \overline{)x^3-a^3}\end{array}$ .

Multiply $x^2$ by $x-a$

  $\begin{array}{l}=x^2\left(x-a\right)\\ =x^3-a{x}^2\end{array}$

Enter the result below the dividend.

  $x-a\begin{array}{c}\hfill x^2\\ \hfill \overline{)\begin{array}{l}{x}^3-a^3\\ x^3-a{x}^2\end{array}}\end{array}$ .

Subtract $x^3-a{x}^2$ from $x^3-a^3$ by changing sign of each term of $x^3-a{x}^2$ and write the remaining term below.

  $x-a\begin{array}{c}\hfill x^2\\ \hfill \overline{)\begin{array}{l}{x}^3-a^3\\ \underset{\_}{x^3-a{x}^2}\\ a{x}^2-a^3\end{array}}\end{array}$ .

Repeat the previous step by taking $a{x}^2-a^3$ .

  $x-a\begin{array}{c}\hfill x^2+ax\\ \hfill \overline{)\begin{array}{l}{x}^3-a^3\\ \underset{\_}{x^3-a{x}^2}\\ a{x}^2-a^3\\ \underset{\_}{a{x}^2-a^{2}x}\\ a^{2}x-a^3\end{array}}\end{array}$ .

Further repeat the previous steps by taking $a^{2}x-a^3$ as dividend.

  $x-a\begin{array}{c}\hfill x^2+ax+a^2\\ \hfill \overline{)\begin{array}{l}{x}^3-a^3\\ \underset{\_}{x^3-a{x}^2}\\ a{x}^2-a^3\\ \underset{\_}{a{x}^2-a^{2}x}\\ a^{2}x-a^3\\ \underset{\_}{a^{2}x-a^3}\\ 0\end{array}}\end{array}$ .

The quotient is $x^2+ax+a^2$ and remainder is $0$ .

To verify the answer write the relation between dividend, divisor, quotient and remainder

  $\left(\text{Quotient}\right)\times \left(\text{Divisor}\right)+\text{Remainder}=\text{Dividend}$ .

Put the values of quotient, divisor, remainder and dividend in the above equation and simplify

  $\begin{array}{c}\left(x^2+ax+a^2\right)\times \left(x-a\right)+0=x^3-a^3\\ x\left(x^2+ax+a^2\right)-a\left(x^2+ax+a^2\right)=x^3-a^3\\ x^3+a{x}^2+a^{2}x-a{x}^2-a^{2}x-a^3=x^3-a^3\\ x^3-a^3=x^3-a^3\end{array}$ .

The result is verified

Therefore, the quotient is $x^2+ax+a^2$ and remainder is $0$ .