Question

A quadratic polynomial when divided by $(x+2)$ leaves a remainder $1$, and when divided by $(x-1)$, leaves a remainder $4$. What will be the remainder if it is divided by $(x+2)(x-1)$?

• A. $1$
• B. $4$
• C. $(x-3)$
• D. $(x+3)$

Answer 1

Answer: (D)

Let the quadratic polynomial be denoted as $P\left(x\right)$.
The polynomial when divided by $x+2$ gives a remainder of 1. So, from remainder theorom, $P(-2)=1$.
Similarly, the polynomial when divided by $x-1$ gives a remainder of 4. So, from remainder theorom, $P\left(1\right)=4$.
Now, if $P\left(x\right)$ is divided by the product $(x+2)(x-1)$, the remainder can be at most be a linear function.
We can write $P\left(x\right)=C(x+2)(x-1)+(Ax+B)$, where A, B, and C are constants.
Use $P\left(1\right)=4$ and $P(-2)=1$.
We get two equations: $A+B=1$ and $-2A+B=1$.
Solving, we get $A=1$ and $B=3$. Hence, the remainder is
$Ax+B=x+3$