# How do you use synthetic division and Remainder Theorem to find P(a) if P(x) = x^3 - 8x^2 + 5x - 7 and a = 1?

Oct 19, 2016

$P \left(1\right) = - 9$

#### Explanation:

Set up the synthetic division problem with the coefficients of the polynomial as the dividend and the $\textcolor{red}{1}$ as the divisor.

$\textcolor{red}{1} | \textcolor{w h i t e}{a a} 1 \textcolor{w h i t e}{a a a} - 8 \textcolor{w h i t e}{a a a a} 5 \textcolor{w h i t e}{a a} - 7$
$\textcolor{w h i t e}{a a a a} \downarrow$
$\textcolor{w h i t e}{a a {a}^{2} a} \textcolor{b l u e}{1} \textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a}$Pull down the $\textcolor{b l u e}{1}$

$\textcolor{red}{1} | \textcolor{w h i t e}{a a} 1 \textcolor{w h i t e}{a a a} - 8 \textcolor{w h i t e}{a a a a} 5 \textcolor{w h i t e}{a a} - 7$
$\textcolor{w h i t e}{a a a a} \downarrow \textcolor{w h i t e}{a a a a a} \textcolor{\lim e g r e e n}{1} \textcolor{w h i t e}{a a a a a a a a a a a a a}$Multiply $\textcolor{red}{1} \cdot \textcolor{b l u e}{1}$ and write the product
color(white)(aaa^2a)color(blue)1color(white)(aa^(2)a)color(blue)(-7)color(white)(aaaaaaaaaaaaacolor(limegreen)1 under the 8. Add $- 8 + \textcolor{\lim e g r e e n}{1} = \textcolor{b l u e}{- 7}$

$\textcolor{red}{1} | \textcolor{w h i t e}{a a} 1 \textcolor{w h i t e}{a a a} - 8 \textcolor{w h i t e}{a a a a} 5 \textcolor{w h i t e}{a a} - 7$
$\textcolor{w h i t e}{a a a a} \downarrow \textcolor{w h i t e}{a a a a a} \textcolor{\lim e g r e e n}{1} \textcolor{w h i t e}{{a}^{22}} \textcolor{\lim e g r e e n}{- 7} \textcolor{w h i t e}{a a a a A a}$Multiply $\textcolor{red}{1} \cdot \textcolor{b l u e}{- 7}$ and put the product
$\textcolor{w h i t e}{a a {a}^{2} a} \textcolor{b l u e}{1} \textcolor{w h i t e}{a a a a} \textcolor{b l u e}{- 7} \textcolor{w h i t e}{{a}^{2} a} \textcolor{b l u e}{- 2} \textcolor{w h i t e}{a a a a a a} \textcolor{\lim e g r e e n}{- 7}$ under the $5$. Add $5 + \textcolor{\lim e g r e e n}{- 7} = \textcolor{b l u e}{- 2}$

$\textcolor{red}{1} | \textcolor{w h i t e}{a a} 1 \textcolor{w h i t e}{a a a} - 8 \textcolor{w h i t e}{a a a a} 5 \textcolor{w h i t e}{a a} - 7$
$\textcolor{w h i t e}{a a a a} \downarrow \textcolor{w h i t e}{a a a a a} \textcolor{\lim e g r e e n}{1} \textcolor{w h i t e}{a a} \textcolor{\lim e g r e e n}{- 7} \textcolor{w h i t e}{a a a} \textcolor{\lim e g r e e n}{- 2} \textcolor{w h i t e}{a a}$Repeat multiplying and adding
$\textcolor{w h i t e}{a a {a}^{2} a} \textcolor{b l u e}{1} \textcolor{w h i t e}{a {a}^{2} a} \textcolor{b l u e}{- 7} \textcolor{w h i t e}{{a}^{11}} \textcolor{b l u e}{- 2} \textcolor{w h i t e}{a a a} \textcolor{m a \ge n t a}{- 9}$

Note that the last number or remainder is $\textcolor{m a \ge n t a}{- 9}$.

According to the remainder theorem, $P \left(1\right) = \textcolor{m a \ge n t a}{- 9}$