# How do you use synthetic substitution to find P(2) for  P(x) = 4x^3 - 5x^2 + 7x - 9?

Jun 19, 2015

$P \left(2\right)$ is the remainder when $P \left(x\right)$ is divided by $x - 2$. You can use synthetic division to do that division. (You could also use long division.)

#### Explanation:

I'm still working on finding a good way to format division here, but if you know synthetic division at all, I think this will get the idea across.

$\left.\begin{matrix}1 & \text{|" & 4 & -5 & 7 & -9 \\ color(white)"1" & "|" & color(white)"ss" & 8 & 6 & 26 \\ color(white)"1" & color(white)"1} & 4 & 3 & 13 & 17\end{matrix}\right.$

The quotient is $4 {x}^{2} + 3 x + 13$ and the remainder (which is what we want) is $17$.

So, the Remainder Theorem tells us that $P \left(2\right) = 17$.

It may not seem like a big deal, but for many people this is faster tan evaluating $P \left(2\right)$ by doing:

$4 {\left(2\right)}^{3} - 5 {\left(2\right)}^{2} + 7 \left(2\right) - 9$ to get $17$

And there are other uses of this fact (theorem).