One factor of [tex]f(x)=4 x^3-4 x^2-16 x+16[/tex] is [tex](x-2)[/tex]. What are all the roots of the function? Use the Remainder Theorem.
Answer (1)
Factor the given polynomial f ( x ) by ( x − 2 ) to get f ( x ) = ( x − 2 ) ( 4 x 2 + 4 x − 8 ) .
Simplify the quadratic factor by dividing by 4: x 2 + x − 2 = 0 .
Factor the simplified quadratic: ( x + 2 ) ( x − 1 ) = 0 , which gives roots x = − 2 and x = 1 .
The roots of f ( x ) are x = − 2 , x = 1 , and x = 2 , so the final answer is x = − 2 , x = 1 , x = 2 .
Explanation
Understanding the Problem We are given the polynomial f ( x ) = 4 x 3 − 4 x 2 − 16 x + 16 and told that ( x − 2 ) is a factor. We want to find all the roots of f ( x ) . This means we want to find all values of x such that f ( x ) = 0 .
Polynomial Division Since ( x − 2 ) is a factor of f ( x ) , we can divide f ( x ) by ( x − 2 ) to find the other factor. Performing polynomial division or synthetic division, we have:
f ( x ) = ( x − 2 ) ( 4 x 2 + 4 x − 8 )
Simplifying the Quadratic Now we need to find the roots of the quadratic 4 x 2 + 4 x − 8 . We can simplify this by dividing by 4, giving us x 2 + x − 2 = 0 .
Factoring the Quadratic We can factor the quadratic as ( x + 2 ) ( x − 1 ) = 0 . Thus, the roots of the quadratic are x = − 2 and x = 1 .
Finding All Roots Therefore, the roots of f ( x ) are x = 2 , x = − 2 , and x = 1 .
Final Answer The roots of the function f ( x ) = 4 x 3 − 4 x 2 − 16 x + 16 are x = − 2 , x = 1 , and x = 2 .
Examples
Polynomials are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, or the behavior of electrical circuits. Finding the roots of a polynomial helps us understand the points where the model intersects the x-axis, which can represent key values in the context of the problem. For example, in projectile motion, the roots can represent the time when the projectile hits the ground.
