Question

Let $P(x)$ 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. $2x - 16$
• D. $60$

Answer 1

Answer: (A)

$\because P(x)=(x-3)(x-5)$
$=Q(x)+(a x +b)$
Given, $P(3)=10$ and $P(5)=6$
$\Rightarrow 3 a +b=10$ ...(i)
and $5 a +b=6$ ...(ii)
On solving Eqs. (i) and (ii), we get
$a=-2$ and $b=16$
$\therefore $ Remainder $=-2 x +16$