Question

When $x^4+x^3-2x^2+x+1$ is divided by $x-1$, the remainder is 2 and the 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 $x^3+2x^2+1$

By division algorithm,
$f\left(x\right)=g\left(x\right)\times q\left(x\right)+r\left(x\right)$
where f(x) $\to$ dividend
g(x) $\to$ divisor
q(x) $\to$ quotient
r(x) $\to$ remainder

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

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

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