Question

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

Answer 1

`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`

Answer 2

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.