Question

One factor of f(x)=4x^3-4x^2-16x+16 is (x – 2). What are all the roots of the function? Use the Remainder Theorem.

Answer 1

b.

Explanation:

Answer 2

The roots of f(x) are x=2, 1, -2

Explanation:

Given polynomial is
`f(x)=4x^(3)-4x^(2)-16x+16`

One factor of above polynomial is (x – 2). we have to find all the roots of above polynomial. By applying synthetic division we find the quotient of above polynomial when divided by (x-2).

We get the coefficient
`4x^(2)+4x-8`

Hence, by remainder theorem,

`f(x)=4x^(3)-4x^(2)-16x+16=(x-2)(4x^(2)+4x-8)+0`

=
`(x-2)4(x^(2)+x-2)`

=
`(x-2)(x^(2)+2x-x-2)4`

=
`(x-2)(x(x+2)-1(x+2))4`

=
`(x-2)(x-1)(x+2)4`

To find roots, f(x)=0

⇒
`(x-2)(x-1)(x+2)4=0`

⇒ x=2, 1, -2