$P(x)$ is a polynomial function. If $P(x^2) = (a-b+2)x^3 - 2x^2 + (2a+b+7)x - 20$ , what is $P(a+b)$ ?

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Answer 1 · 2016-06-20T12:13:24

$P(x)$ cannot be a polynomial because $P(x^2)$ would certainly be an even function.

If $P(x) = sum_{i=0}^na_ix^i$ then

$P(x^2) = sum_{i=0}^na_i(x^2)^i = sum_{i=0}^na_i x^{2i}$

The proposition is not feasible once defined $P(x)$ as a polynomial.

Answer 2 · 2016-06-20T22:02:22

$P(a+b) = -12$

$P(x^2) = (a-b+2)x^3-2x^2+(2a+b+7)x-20$

Since $P(x)$ is a polynomial function, any powers of $x$ in $P(x^2)$ must be even, not odd.

So we require $(a-b+2) = 0$ and $(2a+b+7) = 0$

Adding these two equations together, we get: $3a+9 = 0$ , hence $a=-3$ and $b = -1$

$P(x^2) = -2x^2-20$

So:

$P(x) = -2x-20$

Hence:

$P(a+b)$

$= P((-3)+(-1))$

$= P(-4)$

$= -2(-4)-20$

$= 8-20$

$= -12$

P(x)P(x) is a polynomial function. If P(x^2) = (a-b+2)x^3 - 2x^2 + (2a+b+7)x - 20P(x^2) = (a-b+2)x^3 - 2x^2 + (2a+b+7)x - 20 , what is P(a+b)P(a+b) ?