# How do you use synthetic division and the Remainder Theorem to find P(a) P(x) = 2x^3 - x^2 + 10x + 5 ;a = 1/2?

Sep 23, 2015

Divide $P \left(x\right) = 2 {x}^{3} - {x}^{2} + 10 x + 5$ by $x - a$ for $a = \frac{1}{2}$

Here is the synthetic division:
(You could use long division to get the number instead. But your teacher/grader may want to choose the type of division to check your knowledge of it.)

$\frac{1}{2} | 2 \text{ "-1color(white)(XX)10" } \textcolor{w h i t e}{X} 5$
$\textcolor{w h i t e}{1} | \text{ "color(white)(XX1)1" } \textcolor{w h i t e}{X} 0 \textcolor{w h i t e}{X X 1} 5$
" "stackrel("—————————————)
$\textcolor{w h i t e}{1} | 2 \text{ "color(white)(XX)0" "color(white)(1)10" ""|} \textcolor{w h i t e}{1} 10$

The remainder is $10$,
The remainder Theorem says that when we divide a polynomial $P \left(x\right)$ by $x - a$, the remainder is $P \left(a\right)$

When we divided this $P \left(x\right)$ by $x - \frac{1}{2}$ we got remainder $10$.

So, $P \left(\frac{1}{2}\right) = 10$

(Division format from Ernest Z.)