If -1 is a root of [tex]f(x)[/tex], which of the following must be true?
A. A factor of [tex]f(x)[/tex] is [tex](x-1)[/tex].
B. A factor of [tex]f(x)[/tex] is [tex](x+1)[/tex].
C. Both [tex](x-1)[/tex] and [tex](x+1)[/tex] are factors of [tex]f(x)[/tex].
D. Neither [tex](x-1)[/tex] nor [tex](x+1)[/tex] is a factor of [tex]f(x)[/tex].
Answer (1)
If -1 is a root of f(x), then f(-1) = 0.
By the Factor Theorem, if f(-1) = 0, then (x - (-1)) is a factor of f(x).
Simplify (x - (-1)) to (x + 1).
Therefore, (x + 1) is a factor of f(x). The answer is (x+1).
Explanation
Understanding the Problem We are given that -1 is a root of the polynomial f ( x ) . This means that when we substitute x = − 1 into the polynomial, the result is zero, i.e., f ( − 1 ) = 0 . Our goal is to determine which of the given statements must be true based on this information.
Applying the Factor Theorem The Factor Theorem states that if f ( c ) = 0 for some number c , then ( x − c ) is a factor of f ( x ) . In our case, c = − 1 , so we have f ( − 1 ) = 0 . Therefore, ( x − ( − 1 )) must be a factor of f ( x ) .
Simplifying the Factor Simplifying the expression ( x − ( − 1 )) , we get ( x + 1 ) . Thus, ( x + 1 ) is a factor of f ( x ) .
Analyzing the Options Now we examine the given options:
A factor of f ( x ) is ( x − 1 ) . This is not necessarily true, as we only know that ( x + 1 ) is a factor. A factor of f ( x ) is ( x + 1 ) . This must be true, as we have shown. Both ( x − 1 ) and ( x + 1 ) are factors of f ( x ) . This is not necessarily true, as we only know that ( x + 1 ) is a factor. Neither ( x − 1 ) nor ( x + 1 ) is a factor of f ( x ) . This is false, as we know ( x + 1 ) is a factor.
Conclusion Therefore, the statement that must be true is that ( x + 1 ) is a factor of f ( x ) .
Examples
Consider a polynomial representing the trajectory of a ball. If -1 is a root, it means at time -1 (perhaps representing a reference point before the throw), the ball was at a height of 0. Knowing (x+1) is a factor helps us understand and predict the ball's path, which is crucial in sports like baseball or soccer for predicting movements and optimizing strategies. This concept extends to engineering, where understanding roots and factors aids in designing stable and predictable systems.
