How do you use synthetic division to find P(4) for P(x)= x^4 + x^3 + 10x^2 + 9x - 6?

Jul 2, 2015

When $P \left(x\right)$ is divided by $x - 4$, the remainder is $P \left(4\right)$.

Explanation:

(Synthetic Division Formatting by Truong-Son R.)

To find $P \left(4\right)$ by division, divide ${x}^{4} + {x}^{3} + 10 {x}^{2} + 9 x - 6$ by $x - 4$.
The Remainder Theorem to tells us that the remainder when we do the division will be equal to $P \left(4\right)$
Use synthetic division, because we've been told to. (And it is faster than long division.)

First, you let the coefficients of each degree be used in the division ($1 , 1 , 10 , 1 , - 6$).

Then, dividing by $x - 4$ implies that you use $4$ in your upper left. So, draw the bottom and right sides of a square, put $4$ inside it, and then write $\text{1" " 1" " 10" " 9" " -6}$ to the right.

$\text{1} | |$ $\text{ 1" " 1" " 10" " 9" " -6}$
$+$
$\text{ " }$$- - - - -$

First, bring the first $1$ down to the bottom, and multiply it by the $4$. Put that $4$ below the second $1$.

$\text{4} | |$ $\text{1 " " 1" " 10" " 9" " -6}$
$+$ $\text{ " " "" " 4}$
$\text{ " }$$- - - - -$
$\text{ " " " 1}$

$\text{4} | |$ $\text{1 " " 1 " " 10" " 9" " -6}$
$+$ $\text{ " " "" " 4}$
$\text{ " }$$- - - - -$
$\text{ " " " 1 " " 5}$

Multiply $4 \times 5$ and pout the $20$ under the $10$. Then add:

$\text{4} | |$ $\text{ 1 " " 1" " 10" " 9" " -6}$
$+$ $\text{ " " "" " 4 " " 20}$
$\text{ " }$$- - - - - - - -$
$\text{ " " " 1 " " 5" " 30}$

Repeat to get:

$\text{4} | |$ $\text{1 " " 1" " 10" " " " 9" " " " -6}$
$+$ $\text{ " " "" " 4" " 20 " " 120"" " "516}$
$\text{ " }$$- - - - - - - - -$
$\text{ " " " 1 " " 5" " 30 " " 129" " 510}$

The last number on the bottom row is the remainder and is also $P \left(4\right)$, so $P \left(4\right) = 510$