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Question

# Let f(x) be a polynomial function. If f(x) is divided by xâˆ’1,x+1 and x+2, then remainders are 5,3 and 2 respectively. When f(x) is divided by x3+2x2âˆ’xâˆ’2, then remainder is

A
x4
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B
x+4
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C
x2
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D
x+2
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Solution

## The correct option is A x+4We know that when a polynomial f(x) is divided by (x−a), the remainder is f(a)Let f(x) be the polynomial, q(x) be the quotient and r(x) be the remainder.Degree of remainder is always less than divisor. So, let r(x)=ax2+bx+cGiven that f(1)=5; f(−1)=3; f(−2)=2⇒x3+2x2−x−2=(x−1)(x+1)(x+2)⇒f(x)=q(x)×(x−1)(x+1)(x+2)+r(x)⇒f(1)=q(1)×0+r(1)⇒5=r(1) ..(1)⇒f(−1)=q(−1)×0+r(−1)⇒3=r(−1) ..(2)⇒f(−2)=q(−2)×0+r(−2)⇒2=r(−2) ..(3)Substituting these values from (1),(2),(3) in r(x) we get⇒r(x)=ax2+bx+c⇒r(1)=a+b+c=5⇒r(−1)=a−b+c=3⇒r(−2)=4a−2b+c=2Solving for a,b,c we get⇒a=0;b=1;c=4Therefore, r(x)=x+4

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