Question

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

• A. 20
• B. 19
• C. 17
• D. 15

Answer 1

Answer: (B)

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