Question

What is the quotient and the remainder when `x^4 — 2x^2 +3x - 1` is divided by `x+1`?

Answer 1

`x^3 -x^2 -x +4" " rem -5`

Explanation:

`f(x) = x^4 + 0x^3 -2x^2 +3x -1`

FOr the remainder: find `f(-1)`

`f(-1) = (-1)^4 + 0(-1)^3 -2(-1)^2 +3(-1) -1 = -5`

The remainder is -5.

By synthetic division:

Write the coefficients of the expression - include 0 for `x^3`

"the format does not work so well - (follow the method below)
Apart from the first 1, all the numbers in the last row have to be multiplied by -1"

`"1 0 -2 +3 -1"`
`" -1 +1 1 -4 "`

`"1 -1 -1 4 -5"`

Bring down the first 1.
Multiply it by-1 and write the answer under the second number (0).
Add, to get -1.
Multiply -1 by-1 and write the answer under the third number (-2)
Add to get -1
Multiply -1 by-1 and write the answer under the fourth number (3)
Add to get 4.
Multiply 4 by-1 and write the answer under the fifth number (-1)

Add to get -5. This is the remainder.

Each of the numbers in the last row is a coefficient of the terms in descending powers of `x`.

Quotient is: `x^3 -x^2 -x +4" " rem -5`