#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#