Complete the sentence below.
If $f$ is a polynomial function and $x-5$ is a factor of $f$, then $f(5)=$ ______.
Answer (1)
If x − 5 is a factor of the polynomial f ( x ) , then:
The factor theorem states that f ( 5 ) = 0 .
This is because f ( x ) = ( x − 5 ) e q g ( x ) for some polynomial g ( x ) .
Evaluating f ( 5 ) gives f ( 5 ) = ( 5 − 5 ) e q g ( 5 ) = 0 .
Therefore, f ( 5 ) = 0 .
Explanation
Understanding the Problem We are given that f is a polynomial function and x − 5 is a factor of f . We need to find the value of f ( 5 ) . The factor theorem states that if x − a is a factor of a polynomial f ( x ) , then f ( a ) = 0 . In this case, x − 5 is a factor of f ( x ) , so we can apply the factor theorem with a = 5 .
Applying the Factor Theorem According to the factor theorem, if x − 5 is a factor of f ( x ) , then f ( 5 ) = 0 . This is because if x − 5 is a factor, then f ( x ) can be written as f ( x ) = ( x − 5 ) e q g ( x ) , where g ( x ) is another polynomial. When we evaluate f ( 5 ) , we get f ( 5 ) = ( 5 − 5 ) e q g ( 5 ) = 0 e q g ( 5 ) = 0 .
Conclusion Therefore, f ( 5 ) = 0 .
Examples
The factor theorem is a fundamental concept in algebra that helps us understand the relationship between the roots of a polynomial and its factors. For example, if we know that a polynomial f ( x ) has a factor of ( x − 2 ) , then we know that f ( 2 ) = 0 . This can be useful in finding the roots of polynomials or in simplifying algebraic expressions. Imagine you are designing a bridge and need to ensure that a certain support can handle a specific load. You can model the load as a polynomial function, and if you know a certain value that makes the function zero, you can ensure the support is strong enough at that critical point. This ensures safety and stability in your design.
