How do you divide (x^4 - x^3 - 38x^2 - 31x + 45) -: (x+5), using synthetic division?

Mar 10, 2018

$\frac{{x}^{4} - {x}^{3} - 38 {x}^{2} - 31 x + 45}{x + 5} = {x}^{3} - 6 {x}^{2} - 8 x + 9$

Explanation:

Given: Use synthetic division to divide: $\left({x}^{4} - {x}^{3} - 38 {x}^{2} - 31 x + 45\right) \div \left(x + 5\right)$

In synthetic division the divisor must be a linear factor: $\left(x - a\right)$
The value used to divide is x - a = 0; " "x = a

In synthetic division a coefficient is required for every term. This means if there is a missing term such as $0 {x}^{3}$, a zero would be required.

First list the $x$-value, then each coefficient. Drop down the leading coefficient to the 3rd row (sum row).

$\underline{- 5 |} \text{ " 1" " -1" " -38" " -31" } 45$
" "ul(+" ")
$\text{ } 1$

Multiply the sum in the first column by the $x$-value and place in the second row. Add the first and 2nd row and place the sum in the 3rd row:
$\underline{- 5 |} \text{ " 1" " -1" " -38" " -31" } 45$
" "ul(+" "-5)
$\text{ "1" } - 6$

Multiply the sum in the 2nd column by the $x$-value and place in the 3rd row. Add the first and 2nd row and place the sum in the 3rd row:
$\underline{- 5 |} \text{ " 1" " -1" " -38" " -31" } 45$
" "ul(+" "-5" "30)
$\text{ "1" "-6" } - 8$

Continue this process:
$\underline{- 5 |} \text{ " 1" " -1" " -38" " -31" } 45$
" "ul(+" "-5" "30" "40" "-45)
$\text{ "1" "-6" "-8" "9" } 0$

#The 3rd row represents the coefficients of the quotient. They will always be one less degree than the original equation. The last number is the remainder:

$\frac{{x}^{4} - {x}^{3} - 38 {x}^{2} - 31 x + 45}{x + 5} = {x}^{3} - 6 {x}^{2} - 8 x + 9$