Question

When $f\left(x\right)=2x+q$ is divided by $x-2$, it leaves a remainder of 2. Then, what is the remainder it leaves when divided by $x-3$?

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

The correct option is C 4

Remainder theorem states that if $p\left(x\right)$ is any polynomial of degree greater than or equal to one and $a$ is any real number, then the remainder obtained when $p\left(x\right)$ is divided by the linear polynomial $x-a$ is $p\left(a\right)$.

Given, $f\left(x\right)=2x+q$ and $f\left(2\right)=2$.
$f\left(2\right)=2\Rightarrow 2\left(2\right)+q=2$
$\Rightarrow q=-2$
$\therefore$ $f\left(x\right)=2x-2$

Now, by remainder theorem, we know that when $f\left(x\right)=2x-2$ is divided by $x-3$, the remainder obtained is $f\left(3\right)$.
And, $f\left(3\right)=2\left(3\right)-2=4$.
Hence, the remainder obtained when $f\left(x\right)=2x+q$ is divided by $x-3$ is 4.