Question

Let $f\left(x\right)$ be polynomial in $x$ of degree not less than $1$ and ${}^{\prime }{a}^{\prime }$ be a real number. If $f\left(x\right)$ is divided by $(x-a)$, then the remainder is $f\left(a\right)$. If $(x-a)$ is a factor of $f\left(x\right)$, then $f\left(a\right)=0$. Find the remainder of $x^4+x^3-x^2+2x+3$ when divided by $x-3.$

• A. $108$
• B. $98$
• C. $165$
• D. $170$

Answer 1

The correct option is A $108$

Let,
$f\left(x\right)=x^4+x^3-x^2+2x+3$ and
$g\left(x\right)=x-3$
$=>x-3=0$
$=>x=3$
If $f\left(x\right)$is divided by $g\left(x\right)$, then $f\left(3\right)$ is the remainder(By remainder Theorem)
Thus,
$f\left(3\right)=3^4+3^3-3^2+2\left(3\right)+3$
$f\left(3\right)=81+27-9+6+3$
$f\left(3\right)=108$