Question

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

Answer 1

`(x^4-x^3-38x^2-31x+45)/(x+5) = x^3 -6x^2-8x+9`

Explanation:

Given: Use synthetic division to divide: `(x^4-x^3-38x^2-31x+45)-:(x+5)`

In synthetic division the divisor must be a linear factor: `(x - a)`
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).

`ul(-5|)" " 1" " -1" " -38" " -31" " 45`
`" "ul(+" ")`
`" "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:
`ul(-5|)" " 1" " -1" " -38" " -31" " 45`
`" "ul(+" "-5)`
`" "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:
`ul(-5|)" " 1" " -1" " -38" " -31" " 45`
`" "ul(+" "-5" "30)`
`" "1" "-6" "-8`

Continue this process:
`ul(-5|)" " 1" " -1" " -38" " -31" " 45`
`" "ul(+" "-5" "30" "40" "-45)`
`" "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:

`(x^4-x^3-38x^2-31x+45)/(x+5) = x^3 -6x^2-8x+9`