Finding The Roots: A Math Mystery

Hey math enthusiasts! Let's dive into a classic problem that often pops up: determining the roots of a function. This isn't just about finding a root; it's about understanding the relationships between roots, especially when we're dealing with complex numbers. The question, "Which of the following must also be a root of the function?", forces us to consider some key mathematical concepts, so let's break it down, shall we? We will explore how to find the roots and find out the answer among the options: A. -3, B. -5, C. 2-i, D. 2i.

Understanding Roots and Polynomial Functions

Alright, first things first, what exactly is a root? In the simplest terms, a root of a function (often a polynomial) is a value of the variable (usually 'x') that makes the function equal to zero. Picture this: you've got a function, and you plug in a number. If the function's output is zero, boom, you've found a root! Graphically, these roots are where the function's graph crosses the x-axis. Now, when we talk about polynomials, things get a bit more interesting. Polynomials are expressions with variables raised to non-negative integer powers (like x², x³, etc.) with coefficients (the numbers in front of the variables). The degree of the polynomial (the highest power of the variable) tells us the maximum number of roots the function can have. For example, a quadratic equation (degree 2) can have up to two roots. It is important to know about the Fundamental Theorem of Algebra, which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem also implies that a polynomial of degree n has exactly n complex roots, counted with multiplicity. So, if a root repeats (e.g., if x-2 is a factor twice), we count it twice.

Now, here's where things get juicy. Polynomial functions with real coefficients (meaning the coefficients are real numbers) have a special property: complex roots always come in conjugate pairs. What does this mean, you ask? Well, if a complex number (like a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, the square root of -1) is a root, then its conjugate (a - bi) is also a root. This is the cornerstone of solving the problem in question. This conjugate pair property is super important to remember. It helps us find missing roots efficiently. It's like having a secret code to unlock the puzzle of polynomial functions! This concept is crucial when solving for the missing root of the function.

Now, let's look at the given options to see which must also be a root, considering the context of polynomial functions and complex conjugates.

Analyzing the Options and Finding the Answer

Okay, let's dissect the options one by one, with our knowledge of roots and complex conjugates in mind. We are trying to find the missing root and the relationship between complex roots and conjugate pairs.

• Option A: -3

  If -3 were a root, we don't necessarily have a conjugate pair to consider here, as -3 is a real number. Real number roots don't automatically imply another specific root unless we have additional information (like the polynomial having a repeated root). So, -3 on its own doesn't force another specific root.

• Option B: -5

  Similar to option A, if -5 is a root, it is a real number root. It doesn't trigger the conjugate pair rule. Therefore, we can't definitively say that another specific number must also be a root based on -5 alone.

• Option C: 2 - i

  Aha! This is a complex number. If 2 - i is a root, and assuming our polynomial has real coefficients, then its conjugate must also be a root. The conjugate of 2 - i is 2 + i. So, if 2 - i is a root, 2 + i must also be a root. This is a solid contender!

• Option D: 2i

  If 2i is a root, the conjugate is -2i. While -2i might also be a root (depending on the specific function), it's not a direct consequence of 2i alone being a root. We need to find the conjugate pair of the root to determine the answer.

So, based on our understanding of conjugate pairs, the correct answer is C. 2 + i. If 2 - i is a root, then its complex conjugate, 2 + i, must also be a root if the polynomial has real coefficients. This is the key takeaway! Remember that conjugate pairs are only guaranteed when the coefficients of the polynomial are real numbers. This property is what lets us confidently deduce the existence of another root. This approach will work in any scenario where you are asked to determine the conjugate root.

The Conjugate Pair Theorem: A Deep Dive

Let's delve a bit deeper into the Conjugate Root Theorem, which is the underlying principle here. This theorem is a special case of the Complex Conjugate Root Theorem. The theorem is a powerful tool in algebra, especially when dealing with polynomials. To refresh your memory, the theorem states: if a polynomial equation with real coefficients has a complex root a + bi, then its conjugate a - bi is also a root. This is super useful because it allows us to find roots in pairs, which helps us to factorize polynomials and solve equations more effectively. Remember that the coefficients must be real. If the coefficients are not real, then the conjugate pair property does not necessarily hold. If the function has imaginary coefficients, then the conjugate pair theorem does not apply.

Why does this happen? The beauty lies in how complex conjugates behave when plugged into a polynomial with real coefficients. When you substitute a complex number and its conjugate into the polynomial, the imaginary parts work in such a way that they