Question

One factor of f (x ) = 4 x cubed minus 4 x squared minus 16 x 16 is (x – 2). What are all the roots of the function? Use the Remainder Theorem. X = 1, x = 2, or x = 4 x = –2, x = 1, or x = 2 x = 2, x = 4, or x = 16 x = –16, x = 2, or x = 16.

Answer 1

To solve the problem we must know about the Remainder Theorem.

What is the Remainder theorem?

According to the remainder theorem, when a polynomial P(x) is divided by (x-t) then the remainder of the division is equal to P(t). If P(t)=0, then the (x-t) is the factor of the polynomial.

The roots of the function are 2, 1, and -2.

Given to us

• One factor of f (x) = $4x^3-4x^2-16x+16$ is (x – 2).

What is the quotient of the function?

We know (x-2) is the factor of the function, f(x) = $4x^3-4x^2-16x+16$,

therefore,

$f(x) =4x^3-4x^2-16x+16 = [(x-2) \times quotient] + Remainder$

As (x-2) is the factor of the function, therefore, the remainder will be zero for the equation,

$\rm quotient=\dfrac{4x^3-4x^2-16x+16}{(x-2)}$

$\rm quotient=4x^2+4x-8$

What are the factors of the function?

Solving the quadratic equation,

$4x^2+4x-8=0\\\text{Dividing botht the sides of the equation by 4}\\4x^2+4x-8=0\\(x-1)(x+2)=0$

Hence, the roots of the function are 2, 1, and -2.

Learn more about Remainder theorem:

brainly.com/question/4515216