Root Of F(x): Which Factor Must Be True?

Hey guys! Let's dive into a cool math problem today. We're going to explore what it means for a number to be a root of a polynomial function and how that relates to the factors of the polynomial. This is a fundamental concept in algebra, and understanding it will help you tackle all sorts of polynomial problems. So, grab your thinking caps, and let's get started!

The problem we're tackling today revolves around the relationship between the roots of a polynomial function, $f(x)$, and its factors. Specifically, we're given that -1 is a root of $f(x)$, and we need to determine which of the given statements must be true. This involves understanding the Factor Theorem, which is a cornerstone concept in polynomial algebra. The Factor Theorem essentially links the roots of a polynomial to its linear factors. To truly grasp this, we'll need to not only understand the theorem itself but also how it applies in practice. We'll walk through the theorem step by step, illustrating it with examples to make sure it sticks. Think of it this way: if you know a root, you can figure out a factor, and vice versa. It's like having a secret code to unlock polynomial equations!

Understanding the Factor Theorem isn't just about solving this specific problem; it's a foundational skill that you'll use again and again in algebra and beyond. It helps in simplifying polynomials, finding solutions to equations, and even in graphing functions. Imagine you're trying to find the zeros of a complex polynomial – the Factor Theorem is your trusty tool! So, we'll break down the theorem in a way that’s easy to remember and apply. We’ll look at why it works, how to use it, and common pitfalls to avoid. By the end of this, you'll be able to confidently identify factors from roots and vice versa, making polynomial problems way less daunting. Trust me, this knowledge will be your superpower in the world of algebra!

Let's break down the problem. We are given that $-1$ is a root of the polynomial $f(x)$. What does this actually mean? Well, a root of a function is a value that, when plugged into the function, makes the function equal to zero. In mathematical terms, if $-1$ is a root of $f(x)$, then $f(-1) = 0$. This is a crucial piece of information because it connects the value $-1$ to the function $f(x)$ in a very specific way. It's like saying $-1$ is a special key that unlocks a zero value from the function. This relationship is what the Factor Theorem builds upon, so understanding this basic definition is super important.

Now, we have three options to consider:

A. A factor of $f(x)$ is $(x - 1)$. B. A factor of $f(x)$ is $(x + 1)$. C. Both $(x - 1)$ and $(x + 1)$ are factors of $f(x)$.

These options are all statements about what must be true given that $-1$ is a root. This