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) ?

2 Answers
Jun 20, 2016

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

Explanation:

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.

Jun 20, 2016

P(a+b) = -12

Explanation:

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