If $(x+8)$ is a factor of $f(x)$, which of the following must be true?
A. Both $x=-8$ and $x=8$ are roots of $f(x)$.
B. Neither $x=-8$ nor $x=8$ is a root of $f(x)$.
C. $f(-8)=0$
D. $f(8)=0
Answer (1)
If ( x + 8 ) is a factor of f ( x ) , then by the factor theorem, f ( − 8 ) = 0 .
The factor theorem states that if ( x − a ) is a factor of f ( x ) , then f ( a ) = 0 .
Since ( x + 8 ) = ( x − ( − 8 )) , we have a = − 8 .
Therefore, f ( − 8 ) = 0 , so the correct answer is f ( − 8 ) = 0 .
Explanation
Understanding the Problem We are given that ( x + 8 ) is a factor of a function f ( x ) . We need to determine which of the given statements must be true.
Applying the Factor Theorem The factor theorem states that if ( x − a ) is a factor of a polynomial f ( x ) , then f ( a ) = 0 . In our case, the factor is ( x + 8 ) , which can be written as ( x − ( − 8 )) . Therefore, a = − 8 .
Determining the Correct Statement According to the factor theorem, since ( x + 8 ) is a factor of f ( x ) , it must be true that f ( − 8 ) = 0 . This means that x = − 8 is a root of the function f ( x ) .
Conclusion Now, let's examine the given options:
Both x = − 8 and x = 8 are roots of f ( x ) . This is not necessarily true, as we only know that x = − 8 is a root.
Neither x = − 8 nor x = 8 is a root of f ( x ) . This is false because we know x = − 8 is a root.
f ( − 8 ) = 0 . This is true based on the factor theorem.
f ( 8 ) = 0 . This is not necessarily true, as we have no information about f ( 8 ) .
Therefore, the correct statement is f ( − 8 ) = 0 .
Examples
The factor theorem is used extensively in polynomial algebra and calculus. For example, if you are designing a bridge and model the load distribution as a polynomial function, knowing the roots (where the function equals zero) helps you identify critical points where the structure might be under stress. Similarly, in signal processing, roots of polynomials can represent frequencies that need to be filtered out to reduce noise. Understanding the relationship between factors and roots is crucial for analyzing and manipulating these models effectively.
