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) must be a factor of f(x). ( x + 1 )
Explanation
Understanding the Problem If -1 is a root of f ( x ) , it means that when we substitute x = − 1 into the function, the result is zero, i.e., f ( − 1 ) = 0 . This is a key piece of information that we'll use to determine which of the given statements must be true.
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 Now, let's simplify the expression ( x − ( − 1 )) . Since subtracting a negative number is the same as adding its positive counterpart, we have ( x − ( − 1 )) = ( x + 1 ) . Thus, ( x + 1 ) is a factor of f ( x ) .
Conclusion Based on our analysis, we can conclude that a factor of f ( x ) must be ( x + 1 ) . The other options are not necessarily true. For example, ( x − 1 ) is not necessarily a factor of f ( x ) , and it's not necessary that both ( x − 1 ) and ( x + 1 ) are factors.
Examples
Understanding roots and factors of polynomials is crucial in many areas, such as engineering and physics. For example, when designing a bridge, engineers need to find the roots of certain equations to ensure the structure's stability. If a polynomial equation describes the forces acting on the bridge, finding its roots helps identify critical points where the forces are balanced or reach extreme values. Similarly, in physics, finding the roots of equations can help determine the equilibrium points of a system or the resonant frequencies of a circuit. Knowing that if -1 is a root of a polynomial f(x), then (x+1) is a factor of f(x) allows engineers and physicists to simplify equations and solve problems more efficiently.
