Question

Find the remainder when $x^3+3x^2+3x+1$ is divided by $x$.

• A. $1$
• B. $2$
• C. $\frac{1}{2}$
• D. $\frac{3}{2}$

Answer 1

Answer: (A)

$\left(1\right)$

Given polynomial $x^3+3x^2+3x+1$ needs to be divided by a linear polynomial $x$.
The polynomial remainder theorem states that the remainder of the division of a polynomial $f\left(x\right)$ by linear factor $(x-a)$ is equal to $f\left(a\right)$.
So, let $f\left(x\right)=x^3+3x^2+3x+1$ and the linear factor $x=x-0$ compare with $x-a$ then $a=0$.
Thus, remainder $=f\left(0\right)=(0{)}^3+3(0{)}^2+3\left(0\right)+1$
$\Rightarrow f\left(0\right)=0+0+0+1$
$\therefore f\left(0\right)=1$