Question

$\text{When}{x}^4+x^3-2x^2+x+1\text{is divided}\text{by x-1, the remainder is 2 and the}\text{quotient is q(x). Find q(x).}$

• A. $x^3+2x^2+x+1$
• B. $x^3+x^2+x$
• C. $x^3+2x^2+1$
• D. $x^3+x^2+1$

Answer 1

The correct option is C $x^3+2x^2+1$

$\text{Given:}\text{Dividend =}{x}^4+x^3-2x^2+x+1\text{Divisor =}x-1\text{Remainder = 2}$

$\text{We know that, by division algorithm,}$
$\text{dividend = divisor}\times \text{quotient}$$\text{+ remainder.}$

$\Rightarrow x^4+x^3-2x^2+x+1=(x–1)q\left(x\right)+2$

$x^4+x^3-2x^2+x-1=(x-1)q\left(x\right)$

$q\left(x\right)=\frac{x^4+x^3-2x^2+x-1}{(x–1)}$

$$\begin{array}{c}{x}^3+2x^2+1\\ x-1x^4+x^3-2x^2+x-1\\ \underset{–}{x^4-x^3}2\\ 2x^3-2x^2+x-1\\ \underset{–}{2x^3-2x^2}\\ x-1\\ \underset{–}{x-1}\\ 0\end{array}$$
$\text{So,}q\left(x\right)=x^3+2x^2+1.$