Question

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If a polynomial p(x) having a degree of 5 such that $x^{2}$ is having a coefficient of 1. Also, p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 4 , p(5) = 5, find p(6).

Answer 1

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Answer 2

Given P(1)=1,P(2)=2,P(3)=3,P(4)=5

Let f(x)=(P(x)−x)

f(1)=P(1)−1=1−1=0

f(2)=P(2)−2=2−2=0

f(3)=P(3)−3=3−3=0

∴f(x)=0,x=1,2,3

⇒f(x)=a(x−1)(x−2)(x−3)

P(x)=a(x−1)(x−2)(x−3)+x

Put x=4

5=a(3)(2)(1)+4

⇒a= 1/6

∴P(x)= 1/6 (x−1)(x−2)(x−3)+x

∴P(6)= 1/6 (5)(4)(3)+6=16