If a polynomial function [tex]$f(x)$[/tex] has roots [tex]$3+\sqrt{5}$[/tex] and -6, what must be a factor of [tex]$f(x)$[/tex]?
A. [tex]$(x+(3-\sqrt{5})$[/tex]
B. [tex]$(x-(3-\sqrt{5}))$[/tex]
C. [tex]$(x+(5+\sqrt{3}))$[/tex]
D. [tex]$(x-(5-\sqrt{3}))$[/tex]
Answer (1)
If r is a root of a polynomial f ( x ) , then ( x − r ) is a factor of f ( x ) .
Since 3 + 5 is a root, its conjugate 3 − 5 is also a root.
Therefore, ( x − ( 3 − 5 )) must be a factor of f ( x ) .
The correct answer is ( x − ( 3 − 5 )) .
Explanation
Understanding the Problem We are given that a polynomial function f ( x ) has roots 3 + 5 and − 6 . We need to find a factor of f ( x ) from the given options.
Finding Factors from Roots If r is a root of a polynomial f ( x ) , then ( x − r ) is a factor of f ( x ) . Since 3 + 5 is a root, then ( x − ( 3 + 5 )) is a factor. Since − 6 is a root, then ( x − ( − 6 )) = ( x + 6 ) is a factor. Also, since the coefficients of the polynomial are real, if 3 + 5 is a root, then its conjugate 3 − 5 is also a root. Thus, ( x − ( 3 − 5 )) is a factor.
Checking the Options Now, let's check the given options:
( x + ( 3 − 5 )) : This is not a factor since the root would be − ( 3 − 5 ) = − 3 + 5 , which is not given.
( x − ( 3 − 5 )) : This is a factor since 3 − 5 is a root (conjugate of 3 + 5 ).
( x + ( 5 + 3 )) : This is not a factor since the root would be − ( 5 + 3 ) , which is not given.
( x − ( 5 − 3 )) : This is not a factor since the root would be 5 − 3 , which is not given.
Final Answer Therefore, the correct factor is ( x − ( 3 − 5 )) .
Examples
Polynomial functions and their roots are fundamental in many areas of mathematics and engineering. For example, in control systems, the roots of the characteristic equation determine the stability of the system. If a system has a root with a positive real part, the system is unstable. Similarly, in signal processing, the roots of a polynomial can represent the frequencies present in a signal. Knowing the roots helps in designing filters to remove unwanted frequencies. In structural engineering, understanding the roots of equations helps in analyzing the vibrational modes of a structure, ensuring it can withstand external forces without collapsing.
