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)
The problem states that ( x + 8 ) is a factor of f ( x ) .
The factor theorem states that if ( x − a ) is a factor of f ( x ) , then f ( a ) = 0 .
Applying the factor theorem, since ( x + 8 ) is a factor, f ( − 8 ) = 0 .
Therefore, the correct answer is f ( − 8 ) = 0 .
Explanation
Understanding the Problem We are given that ( x + 8 ) is a factor of f ( x ) . We need to determine which of the given statements must be true.
Recalling the Factor Theorem Recall the factor theorem: If ( x − a ) is a factor of f ( x ) , then f ( a ) = 0 .
Applying the Factor Theorem In this case, we have the factor ( x + 8 ) , which can be written as ( x − ( − 8 )) . Therefore, a = − 8 .
Determining the Correct Statement According to the factor theorem, if ( x + 8 ) is a factor of f ( x ) , then f ( − 8 ) = 0 .
Checking the Options Now we check the given options to see which one matches this conclusion:
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, as 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.
Final Answer Therefore, the correct statement is f ( − 8 ) = 0 .
Examples
The factor theorem is a fundamental concept in algebra that links the factors of a polynomial to its roots. For example, if you are designing a bridge and model the load distribution as a polynomial function, knowing the roots (where the load is zero) can help you identify critical support points. Similarly, in cryptography, understanding polynomial factorization is crucial for designing secure encryption algorithms. This theorem provides a powerful tool for analyzing and solving real-world problems involving polynomials.
