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Question
A polynomial p(x) when divided by (x−3), leaves a remainder of 5. But when p(x) is divided by (x−5), the remainder is 3. Let r(x) be the remainder when p(x) is divided by (x−3)(x−5). Then the value of r(1) is
A polynomial p(x) when divided by (x−3), leaves a remainder of 5. But when p(x) is divided by (x−5), the remainder is 3. Let r(x) be the remainder when p(x) is divided by (x−3)(x−5). Then the value of r(1) is
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Solution
The correct option is D 7
By remainder theorem,
p(3)=5 and p(5)=3
Let p(x)=(x−3)(x−5)q(x)+r(x)
Since the degree of remainder is always less than the degree of divisor, we get
r(x)=Ax+B
⇒3A+B=5 and 5A+B=3
⇒A=−1 and B=8
∴r(x)=−x+8
⇒r(1)=7
By remainder theorem,
p(3)=5 and p(5)=3
Let p(x)=(x−3)(x−5)q(x)+r(x)
Since the degree of remainder is always less than the degree of divisor, we get
r(x)=Ax+B
⇒3A+B=5 and 5A+B=3
⇒A=−1 and B=8
∴r(x)=−x+8
⇒r(1)=7
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