When a polynomial p (x) is divided by (x-1), the remainder is 5, and when it is divided by (x-2), the remainder is 7. Find the remainder when f (x) is divided by (x-1)(x-2)

Q. Let p(x) be a polynomial of degree more than 2. When p(x) is divided by x-2, it leaves remainder 1 and when it is divided by x-3, it leaves remainder 3. Find the remainder, when p(x) is divided by (x-2)(x-3).

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When a polynomial p(x) is divided by x−1, the remainder is 3. When p(x) is divided by x−3, the remainder is 5. If r(x) is the remainder when p(x) is divided by (x−1)(x−3), then the value of r(−2) is

The correct option is B 0Given P(1)=3P(3)=5Let r(x)=ax+bSo P(x)=(x−1)(x−3)q(x)+ax+bP(1)=a+b⇒ a+b=3P(3)=3a+b⇒ 3a+b=5 a=1,b=2r(x)=ax+b=x+2r(−2)=(−2)+2)r(−2)=0

When a polynomial $p (x)$ is divided by $x - 1$ , the remainder is $3$ . When $p (x)$ is divided by $x - 3$ , the remainder is $5$ . If $r (x)$ is the remainder when $p (x)$ is divided by $(x - 1) (x - 3)$ , then the value of $r (- 2)$ is

The correct option is B $0$
Given $P (1) = 3$
$P (3) = 5$
Let $r (x) = a x + b$
So $P (x) = (x - 1) (x - 3) q (x) + a x + b$
$P (1) = a + b \Rightarrow$ $a + b = 3$
$P (3) = 3 a + b \Rightarrow$ $3 a + b = 5$
$a = 1$ , $b = 2$
$r (x) = a x + b = x + 2$
$r (- 2) = (- 2) + 2)$
$r (- 2) = 0$