Question

Let $P\left(x\right)$ be a polynomial, which when divided by $x-3$ and $x-5$ leaves remainders $10$ and $6$ respectively. If the polynomial is divided by $(x-3)(x-5)$ then the remainder is

• A. $-2x+16$
• B. $16$
• C. $60$
• D. $2x-16$

Answer 1

The correct option is A $-2x+16$

Let the remainder be, $ax+b$
From the given data,
$P\left(x\right)=(x-3)(x-5)q\left(x\right)+(ax+b)$

When $x=3$, $P\left(3\right)=10$
and when $x=5$, $P\left(5\right)=6$
The remainder is,
$3a+b=10$
$5a+b=6$

Solving the above equations,
$a=-2,b=16$
Thus, the remainder is, $-2x+16$.