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  4. Let P ( x ) be quadratic polynomial with real coefficients such that for all real x the relation 2(1+ P ( x ))= P ( x -1)+ P ( x +1) holds. If P (0)=8 and P (2)=32 then Sum of all the coefficients of P(x) is

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Q. Let $P ( x )$ be quadratic polynomial with real coefficients such that for all real $x$ the relation $2(1+ P ( x ))= P ( x -1)+ P ( x +1)$ holds. If $P (0)=8$ and $P (2)=32$ then
Sum of all the coefficients of $P(x)$ is

Complex Numbers and Quadratic Equations

Solution:

Put $x=1$ in $2(1+P(x))=P(x-1)+P(x+1)$
$2(1+ P (1))= P (0)+ P (2) \Rightarrow 2+2 P (1)=8+32 \Rightarrow 2 P (1)=38 \Rightarrow P (1)=19$
hence sum of all the coefficient is 19