Question

How do you use the remainder theorem and synthetic division to find the remainder when `(x^4-x^3-5x^2-x-6) div (x-3)`?

Answer 1

We use the remainder theorem to determine whether (x-3) is a factor of the expression. The remainder is 0. Therefore it is a factor and we can use synthetic division to find the quotient.
`(x^4 - x^3 -5x^2 -x -6) ÷ (x-3) = x^3 + 2x^2 + x + 2`

Explanation:

`f(x) = x^4 - x^3 -5x^2 -x -6`

The factors of 6 are: 1, -1, 2, -2, 3, -3, 6, -6.
Substitute each into f(x) until you obtain a result of 0.

`f(1) != 0` so `(x-1)` is not a factor
`f(-1) != 0` so `(x+1)` is not a factor
`f(-2) = 0` so `(x+2)` is a factor. etc..

However, we are asked to find the remainder when the expression is divided by `(x-3).` Substitute `x = 3` into the expression.

`f(+3) =3^4 - 3^3 -5(3)^2 -3 -6 = 0`.
This means there is no remainder and (`x-3)` is a factor.

By synthetic division:

Write down only the coefficients of the terms, and write +3 outside.

`3 " ) 1 -1 -5 -1 -6"`
`" 3 6 3 6"`

`" 1 2 1 2 0"`

Bring down the first 1.
Multiply it by 3 and write the answer under the second number. Add.
This gives 2.

Multiply 3 by 2 and write the answer under the third number. Add.
This gives 1.
Multiply 3 by the 1 and write it under the fourth number. Add.
This gives 2.
Multiply 3 by 2 and write it under the fifth number. Add.

This gives 0. Which means there is no remainder and (x-3) is a factor of the expression.

The quotient will start with an `x^3` term because `x^4 ÷ x = x^3`

Use the numbers in last row as the coefficients of the terms with the descending powers of x.

`(x^4 - x^3 -5x^2 -x -6) ÷ (x-3) = x^3 + 2x^2 + x + 2`