Given P(x) = 2x^2-x+2 and Q(x) = 2x-1, what is P(Q(x)) ?

2 Answers
Nov 11, 2017

P(Q(x)) = 8x^2-10x+5

Explanation:

Note that changing the variable name does not really affect what the polynomial is. So we can write:

P(t) = 2t^2-t+2

Then, substituting Q(x) for t it may be clear that:

P(Q(x)) = 2(Q(x))^2-Q(x)+2

color(white)(P(Q(x))) = 2(2x-1)^2-(2x-1)+2

color(white)(P(Q(x))) = 2(4x^2-4x+1)-(2x-1)+2

color(white)(P(Q(x))) = (8x^2-8x+2)-(2x-1)+2

color(white)(P(Q(x))) = 8x^2-10x+5

Nov 11, 2017

Answer: P(Q(x))=8x^2-10x+5

Explanation:

This is a composition of functions problem. Note that P(Q(x)) means to substitute Q(x) as the "x" variable in the P(x) expression.

Therefore, given P(x)=2x^2-x+2 and Q(x)=2x-1:
P(Q(x))=P(2x-1)
=2(2x-1)^2-(2x-1)+2

We can continue simplifying by noting that
(a+-b)^2=a^2+-2ab+b^2

So:
2(2x-1)^2-(2x-1)+2
=2(4x^2-4x+1)-2x+1+2 by the distributive property
=8x^2-8x+2-2x+3 by the distributive property again
=8x^2-10x+5 which is our answer.