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.
A. [tex]x=1, x=2[/tex], or [tex]x=4[/tex]
B. [tex]x=-2, x=1[/tex], or [tex]x=2[/tex]
C. [tex]x=2, x=4[/tex], or [tex]x=16[/tex]
D. [tex]x=-16, x=2[/tex], or [tex]x=16[/tex]
Answer (1)
Perform polynomial division to find the quadratic factor: 4 x 3 − 4 x 2 − 16 x + 16 = ( x − 2 ) ( 4 x 2 + 4 x − 8 ) .
Simplify the quadratic factor: 4 x 2 + 4 x − 8 = 4 ( x 2 + x − 2 ) .
Factor the simplified quadratic: x 2 + x − 2 = ( x + 2 ) ( x − 1 ) .
Identify all roots: x = − 2 , x = 1 , x = 2 .
The roots of the function are x = − 2 , x = 1 , x = 2 .
Explanation
Problem Analysis We are given the polynomial f ( x ) = 4 x 3 − 4 x 2 − 16 x + 16 and told that ( x − 2 ) is a factor. Our goal is to find all the roots of f ( x ) . Since ( x − 2 ) is a factor, we know that x = 2 is one root. We can use polynomial division or synthetic division to find the remaining quadratic factor. Then we can find the roots of the quadratic factor.
Polynomial Division We can perform polynomial division to divide 4 x 3 − 4 x 2 − 16 x + 16 by ( x − 2 ) .
\multicolumn 2 r 4 x 2 \cline 2 − 5 x − 2 \multicolumn 2 r 4 x 3 \cline 2 − 3 \multicolumn 2 r 0 \multicolumn 2 r \cline 3 − 4 \multicolumn 2 r \multicolumn 2 r \cline 4 − 5 \multicolumn 2 r + 4 x 4 x 3 − 8 x 2 4 x 2 4 x 2 0 − 8 − 4 x 2 − 16 x − 8 x − 8 x − 8 x 0 − 16 x + 16 + 16 0 + 16
So, 4 x 3 − 4 x 2 − 16 x + 16 = ( x − 2 ) ( 4 x 2 + 4 x − 8 ) .
Finding Roots of Quadratic Factor Now we need to find the roots of the quadratic 4 x 2 + 4 x − 8 = 0 . We can simplify this by dividing by 4 to get x 2 + x − 2 = 0 . We can factor this quadratic as ( x + 2 ) ( x − 1 ) = 0 . Therefore, the roots are x = − 2 and x = 1 .
Final Roots The roots of f ( x ) are x = 2 , x = − 2 , and x = 1 .
Conclusion Therefore, 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
Understanding the roots of polynomial functions is crucial in many areas of engineering and physics. For example, in structural engineering, finding the roots of a polynomial can help determine the stability of a bridge or building. In physics, roots can represent equilibrium points in a system. By analyzing the polynomial f ( x ) = 4 x 3 − 4 x 2 − 16 x + 16 , we can model a system where the values x = − 2 , 1 , 2 represent critical points that define the behavior of the system. This could represent stable or unstable states, depending on the context.
