# 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)#?

##### 1 Answer

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.

#### Explanation:

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.

However, we are asked to find the remainder when the expression is divided by

This means there is no remainder and (

By synthetic division:

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

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

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