If [tex]f(1)=0[/tex], what are all the roots of the function [tex]f(x)=x^3+3 x^2-x-3[/tex]? Use the Remainder Theorem.
A. [tex]x=-1, x=1[/tex], or [tex]x=3[/tex]
B. [tex]x=-3, x=-1[/tex], or [tex]x=1[/tex]
C. [tex]x=-3[/tex] or [tex]x=1[/tex]
D. [tex]x=-1[/tex] or [tex]x=3[/tex]
Answer (1)
Given f ( x ) = x 3 + 3 x 2 − x − 3 and f ( 1 ) = 0 , recognize x = 1 as a root.
Factor f ( x ) by grouping: f ( x ) = x 2 ( x + 3 ) − ( x + 3 ) = ( x 2 − 1 ) ( x + 3 ) .
Further factor x 2 − 1 as ( x − 1 ) ( x + 1 ) , so f ( x ) = ( x − 1 ) ( x + 1 ) ( x + 3 ) .
Identify the roots by setting each factor to zero: x = 1 , − 1 , − 3 . The roots are x = − 3 , x = − 1 , x = 1 .
Explanation
Understanding the Problem We are given the function f ( x ) = x 3 + 3 x 2 − x − 3 and told that f ( 1 ) = 0 . This means that x = 1 is a root of the function. Our goal is to find all the roots of f ( x ) .
Using the Remainder Theorem Since f ( 1 ) = 0 , by the Remainder Theorem, ( x − 1 ) must be a factor of f ( x ) . We can thus write f ( x ) = ( x − 1 ) q ( x ) , where q ( x ) is a quadratic polynomial. To find q ( x ) , we can perform polynomial division or factor by grouping.
Factoring by Grouping Let's factor by grouping:
f ( x ) = x 3 + 3 x 2 − x − 3 = x 2 ( x + 3 ) − 1 ( x + 3 ) = ( x 2 − 1 ) ( x + 3 )
Factoring Difference of Squares Now, we can further factor ( x 2 − 1 ) as a difference of squares:
x 2 − 1 = ( x − 1 ) ( x + 1 )
Finding the Roots So, we have f ( x ) = ( x − 1 ) ( x + 1 ) ( x + 3 ) . The roots of f ( x ) are the values of x that make f ( x ) = 0 . Thus, the roots are x = 1 , x = − 1 , and x = − 3 .
Final Answer Therefore, the roots of the function f ( x ) = x 3 + 3 x 2 − x − 3 are x = − 3 , x = − 1 , and x = 1 .
Examples
Understanding the roots of a polynomial can help engineers design stable systems. For example, if f ( x ) represents the behavior of a control system, the roots indicate critical points. Knowing these roots ensures the system operates safely and predictably. In circuit analysis, roots of characteristic equations determine the stability of circuits. By finding the roots, engineers can adjust component values to avoid oscillations or instability, ensuring reliable performance.
