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Question: If $f(x)$ is a polynomial of degree $4$ and $f(n)=n+1$ for $n=1,2,3,4$. Find $f(5)$

If we construct $g(x)=f(x)-(x-2)(x-3)(x-4)(x-5)$, then is it possible to find f(5)?

MKS
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  • A polynomial of degree $n$ is determined by $n+1$ points (a line is determined by two points) Since you have been given only four points the fifth point can be arbitrarily chosen, and therefore cannot be determined (see Lagrange interpolation for how, given $n+1$ points a unique polynomial of degree at most $n$ can be found which passes through them). – Mark Bennet Nov 14 '19 at 10:41

2 Answers2

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$f(x)=x+1+c(x-1)(x-2)(x-3)(x-4)$ satisfies the hypothesis for any $c \neq 0$. So it is not possible to find $f(5)$ from the given information.

Kavi Rama Murthy
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Note that, $f(x)=\lambda(x-1)(x-2)(x-3)(x-4)+x+1$. Here $\lambda$ is a non-zero scalar. Hence $f(5)=\lambda 4\cdot 3\cdot 2\cdot 1+6=6+24\lambda$.

Sumanta
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