Chapter A.3, Problem 69AYU

To determine

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

Answer to Problem 69AYU

The quotient is $x^2-x-1$ and remainder is $2x+2$ .

Explanation of Solution

Given information:

  $\text{Divide}\left(1-x^2+x^4\right)\text{by}\left(x^2+x+1\right)$ .

Calculation:

Rearrange the terms of dividend as descending power of variables.

  $x^4-x^2+1$

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

  $x^2+x+1\begin{array}{c}\hfill x^2\\ \hfill \overline{)x^4-x^2+1}\end{array}$ .

Multiply $x^2$ by $x^2+x+1$

  $\begin{array}{l}=x^2\left(x^2+x+1\right)\\ =x^4+x^3+x^2\end{array}$

Enter the result below the dividend.

  $x^2+x+1\begin{array}{c}\hfill x^2\\ \hfill \overline{)\begin{array}{l}{x}^4-x^2+1\\ x^4+x^3+x^2\end{array}}\end{array}$ .

Subtract $x^4+x^3+x^2$ from $x^4-x^2+1$ by changing sign of each term of $x^4+x^3+x^2$ and write the remaining term below.

  $x^2+x+1\begin{array}{c}\hfill x^2\\ \hfill \overline{)\begin{array}{l}{x}^4-x^2+1\\ \underset{\_}{x^4+x^3+x^2}\\ -x^3-2x^2+1\end{array}}\end{array}$ .

Repeat the previous step by taking $-x^3-2x^2+1$ .

  $x^2+x+1\begin{array}{c}\hfill x^2-x\\ \hfill \overline{)\begin{array}{l}{x}^4-x^2+1\\ \underset{\_}{x^4+x^3+x^2}\\ -x^3-2x^2+1\\ \underset{\_}{-x^3-x^2-x}\\ -x^2+x+1\end{array}}\end{array}$ .

Further repeat the previous steps by taking $-x^2+x+1$ as dividend.

  $x^2+x+1\begin{array}{c}\hfill x^2-x-1\\ \hfill \overline{)\begin{array}{l}{x}^4-x^2+1\\ \underset{\_}{x^4+x^3+x^2}\\ -x^3-2x^2+1\\ \underset{\_}{-x^3-x^2-x}\\ -x^2+x+1\\ \underset{\_}{-x^2-x-1}\\ 2x+2\end{array}}\end{array}$ .

Since further division is not possible.

The quotient is $x^2-x-1$ and remainder is $2x+2$ .

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

  $\begin{array}{c}\left(x^2-x-1\right)\times \left(x^2+x+1\right)+2x+2=x^4-x^2+1\\ x^2\left(x^2+x+1\right)-x\left(x^2+x+1\right)-1\left(x^2+x+1\right)+2x+2=x^4-x^2+1\\ x^4+x^3+x^2-x^3-x^2-x-x^2-x-1+2x+2=x^4-x^2+1\\ x^4-x^2+1=x^4-x^2+1\end{array}$ .

The result is verified

Therefore, the quotient is $x^2-x-1$ and remainder is $2x+2$ .