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

Nov 11, 2017

$P \left(Q \left(x\right)\right) = 8 {x}^{2} - 10 x + 5$

#### Explanation:

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

$P \left(t\right) = 2 {t}^{2} - t + 2$

Then, substituting $Q \left(x\right)$ for $t$ it may be clear that:

$P \left(Q \left(x\right)\right) = 2 {\left(Q \left(x\right)\right)}^{2} - Q \left(x\right) + 2$

$\textcolor{w h i t e}{P \left(Q \left(x\right)\right)} = 2 {\left(2 x - 1\right)}^{2} - \left(2 x - 1\right) + 2$

$\textcolor{w h i t e}{P \left(Q \left(x\right)\right)} = 2 \left(4 {x}^{2} - 4 x + 1\right) - \left(2 x - 1\right) + 2$

$\textcolor{w h i t e}{P \left(Q \left(x\right)\right)} = \left(8 {x}^{2} - 8 x + 2\right) - \left(2 x - 1\right) + 2$

$\textcolor{w h i t e}{P \left(Q \left(x\right)\right)} = 8 {x}^{2} - 10 x + 5$

Nov 11, 2017

Answer: $P \left(Q \left(x\right)\right) = 8 {x}^{2} - 10 x + 5$

#### Explanation:

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

Therefore, given $P \left(x\right) = 2 {x}^{2} - x + 2$ and $Q \left(x\right) = 2 x - 1$:
$P \left(Q \left(x\right)\right) = P \left(2 x - 1\right)$
$= 2 {\left(2 x - 1\right)}^{2} - \left(2 x - 1\right) + 2$

We can continue simplifying by noting that
${\left(a \pm b\right)}^{2} = {a}^{2} \pm 2 a b + {b}^{2}$

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