Question

Use the Remainder Theorem to find the remainder for (x^3-x+6) divided by (x-2) and state whether or not the binomial is a factor of the polynomial.

Answer 1

Remainder is 12.

The binomial
`(x-2)` is not a factor of the given polynomial.

Explanation:

The Remainder Theorem states that when a polynomial
`p(x)` is divided by a binomial
`(x-a)`, then the remainder is given as
`p(a)`.

Also, if
`p(a)` is 0, then
`(x-a)` is a factor of the given polynomial.

Here,
`p(x)=x^(3)-x+6` and
`a=2`

So, the remainder on dividing
`p(x)=x^(3)-x+6` by
`(x-2)` is
`p(2)`.

Now,
`p(2)=2^(3)-2+6=8-2+6=12`.

Therefore, the remainder is 12.

∵
`p(2)` is equal to 12 and not 0. So, the binomial
`(x-2)` is not a factor of the given polynomial.