Question

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

Answer 1

When `P(x)` is divided by `x-4`, the remainder is `P(4)`.

Explanation:

(Synthetic Division Formatting by Truong-Son R.)

To find `P(4)` by division, divide `x^4+x^3 + 10x^2 + 9x -6` by `x-4`.
The Remainder Theorem to tells us that the remainder when we do the division will be equal to `P(4)`
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 `"1" " 1" " 10" " 9" " -6"` to the right.

`"1" ||` `" 1" " 1" " 10" " 9" " -6"`
`+`
`"" " "` `-----`

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

`"4" ||` `"1 " " 1" " 10" " 9" " -6"`
`+` `"" " " "" " 4"`
`"" " """` `-----`
`"" " " " 1"`

Then add it up:

`"4" ||` `"1 " " 1 " " 10" " 9" " -6"`
`+` `"" " " "" " 4"`
`"" " "` `-----`
`"" " " " 1 " " 5"`

Multiply `4 xx 5` and pout the `20` under the `10`. Then add:

`"4" ||` `" 1 " " 1" " 10" " 9" " -6"`
`+` `"" " " "" " 4 " " 20"`
`"" " "` `--------`
`"" " " " 1 " " 5" " 30"`

Repeat to get:

`"4" ||` `"1 " " 1" " 10" " " " 9" " " " -6"`
`+` `"" " " "" " 4" " 20 " " 120"" " "516"`
`"" " "` `---------`
`"" " " " 1 " " 5" " 30 " " 129" " 510"`

The last number on the bottom row is the remainder and is also `P(4)`, so `P(4) = 510`