Complex Roots: Finding Polynomial Equation & Other Roots

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Hey guys! Let's dive into a fascinating problem involving polynomial functions and their complex roots. This is a classic topic in algebra, and understanding it can really boost your problem-solving skills. We'll break down the problem step by step, making sure everything is crystal clear.

Understanding the Problem of Polynomial Functions

So, we're given a polynomial function, and the key here is that one of its roots is a complex number, specifically 2 + i. We also know that 5 is another root. The big question is: can we figure out the equation for this polynomial and identify another root from the choices given? This involves understanding a crucial property of polynomials with real coefficients: complex roots always come in conjugate pairs. This principle is fundamental in solving polynomial equations, especially when dealing with non-real solutions.

When dealing with polynomial functions, particularly those with real coefficients, complex roots always occur in conjugate pairs. This is a cornerstone concept when trying to understand the nature and behavior of polynomial equations. In our case, we are told that 2 + i is a root of the polynomial. Knowing this, we can immediately deduce that its complex conjugate must also be a root. The conjugate is formed by simply changing the sign of the imaginary part. So, if 2 + i is a root, then 2 - i is necessarily another root. This is because complex roots arise from irreducible quadratic factors with real coefficients. Think about it – when you solve a quadratic equation that has a negative discriminant (the part under the square root in the quadratic formula), you get complex solutions. These solutions always appear as a pair of conjugates.

Now, let's think about how this affects the construction of the polynomial. Each root corresponds to a factor of the polynomial. If 5 is a root, then (x - 5) is a factor. If 2 + i is a root, then [x - (2 + i)] is a factor. And, crucially, if 2 - i is a root, then [x - (2 - i)] is a factor. To form the polynomial, we multiply these factors together. This ensures that when you plug in any of the roots, the polynomial will evaluate to zero. This multiplication process is where we can really see the magic of complex conjugates at work. When we multiply the factors corresponding to the complex conjugate roots, the imaginary parts tend to cancel out, leaving us with a polynomial that has real coefficients, just as the problem implies. This is a beautiful demonstration of how mathematical structures ensure consistency and harmony in our equations.

Building the Polynomial Equation

Let's start by acknowledging the roots we know: 2 + i and 5. If a polynomial has complex roots, they always come in conjugate pairs. That means if 2 + i is a root, then its conjugate, 2 - i, must also be a root. So, we have three roots: 5, 2 + i, and 2 - i. Remember, each root corresponds to a factor of the polynomial. If 'r' is a root, then (x - r) is a factor. So, our factors are:

  • (x - 5)
  • [x - (2 + i)]
  • [x - (2 - i)]

To form the polynomial, we multiply these factors together. Let's first multiply the factors corresponding to the complex conjugate roots:

[x - (2 + i)] * [x - (2 - i)]

This looks a bit messy, but we can simplify it. Think of it as multiplying two binomials. Expand the expression:

(x - 2 - i) * (x - 2 + i)

= x² - 2x + ix - 2x + 4 - 2i - ix + 2i - i²

Notice something cool? The imaginary terms (+ix and -ix, -2i and +2i) cancel each other out! Also, remember that i² = -1. So, we can simplify further:

= x² - 4x + 4 - (-1)

= x² - 4x + 5

This is a quadratic factor with real coefficients, which is exactly what we expected. Now, we multiply this by the remaining factor (x - 5):

(x² - 4x + 5) * (x - 5)

Expand this:

= x³ - 4x² + 5x - 5x² + 20x - 25

Combine like terms:

= x³ - 9x² + 25x - 25

So, our polynomial function is f(x) = x³ - 9x² + 25x - 25. This is a cubic polynomial (degree 3) with the roots we identified earlier. The process of constructing the polynomial from its roots is a powerful way to see how these roots dictate the shape and behavior of the function. Each root is a value of x that makes the polynomial equal to zero, and the polynomial's factors directly correspond to these roots. Knowing this relationship is incredibly helpful in polynomial manipulation and problem-solving.

Identifying the Conjugate Root

The question also asks us which of the given options must also be a root of the function. We've already touched on this, but let's reiterate the key principle: complex roots of polynomials with real coefficients always come in conjugate pairs. This is a non-negotiable rule in the world of algebra. If you have a polynomial with real number coefficients and you find that a + bi is a root (where 'a' and 'b' are real numbers and 'i' is the imaginary unit), then a - bi must also be a root.

We were given that 2 + i is a root. Therefore, its conjugate, which is 2 - i, must also be a root. Let's look at the options:

  • A. -3
  • B. -5
  • C. 2 - i
  • D. 2i

Clearly, the correct answer is C. 2 - i. The other options don't fit the conjugate pair rule. This highlights the elegance and consistency of complex numbers in polynomial theory. The fact that complex roots always pair up is not just a mathematical curiosity; it’s a fundamental property that helps us understand the structure and solutions of polynomial equations. Without this property, solving and analyzing polynomials with complex roots would be significantly more challenging.

Final Answer

So, to recap, the equation for the polynomial function is f(x) = x³ - 9x² + 25x - 25, and the other root that must also be a root of the function is C. 2 - i. This problem perfectly illustrates the importance of understanding complex conjugates when dealing with polynomial functions. You nailed it!

Remember, the key takeaways here are:

  1. Complex roots come in conjugate pairs.
  2. Each root corresponds to a factor of the polynomial.
  3. Multiplying the factors gives you the polynomial equation.

Keep practicing, and you'll become a pro at these types of problems! You've got this! Understanding these principles not only helps in solving problems but also provides a deeper appreciation for the interconnectedness of mathematical concepts. Polynomials, complex numbers, and their interactions are central to many areas of mathematics and its applications, from engineering to computer science. So, keep exploring and expanding your mathematical horizons!