Question

A quadratic polynomial when divided by $\left( {x + 2} \right)$ leaves a remainder $1$ and when divided by $\left( {x - 1} \right)$ leaves remainder $4$. What will be the remainder if it is divided by $\left( {x + 2} \right)\left( {x - 1} \right)$?
A) $1$
B) $4$
C) $\left( {x + 3} \right)$
D) $\left( {x - 3} \right)$

Answer 1

Hint:We will assume a general expression for the polynomial. The remainders for different divisors are given so we will use the remainder theorem to find out the values of the function at these factors. We will combine both the conditions and then use it to find the required remainder.

Complete step-by-step answer:
Let us consider $f\left( x \right)$ to be the given quadratic polynomial.
It is given that when $f\left( x \right)$ is divided by $\left( {x + 2} \right)$, the remainder is $1$.
Similarly, it is given that when $f\left( x \right)$ is divided by $\left( {x - 1} \right)$, the remainder is $4$.
The remainder theorem states that if any polynomial $p\left( x \right)$ is divided by $\left( {x - a} \right)$ leaving remainder $k$ then $f\left( a \right) = k$ .
So therefore, for the polynomial $f\left( x \right)$, using the given statements we can write that $f\left( { - 2} \right) = 1$ and $f\left( 1 \right) = 4$.
Now let us assume that we are dividing $f\left( x \right)$ by the product $\left( {x + 2} \right)\left( {x - 1} \right)$ then by the division algorithm for the polynomials we can write,
$f\left( x \right) = k\left( {x + 2} \right)\left( {x - 1} \right) + q\left( x \right)$
As it is given that $f\left( x \right)$ is a quadratic polynomial, we conclude that $q\left( x \right)$ is a linear polynomial.
That means that $q\left( x \right) = Ax + B$ .
Therefore,
$f\left( x \right) = k\left( {x + 2} \right)\left( {x - 1} \right) + Ax + B$
We know that $f\left( { - 2} \right) = 1$.
Substituting in the above equation we get the following:
$ - 2A + B = 1$
Similarly, from the condition that $f\left( 1 \right) = 4$ we get,
$A + B = 4$
Solving both the simultaneous equations we get, $A = 1,B = 3$ .
Therefore, the remainder is $x + 3$ .

So, the correct answer is “Option C”.

Note:The important point of this problem is the remainder theorem. We don’t have to find the actual polynomial; we use remainder theorem directly and then use a division algorithm to find the remainder. Just form the proper equation to reach the final answer.