Finding The Polynomial: Roots, Coefficients & Degree

Hey math enthusiasts! Let's dive into a cool problem involving polynomial functions. Our mission? To find the polynomial function of the lowest degree with some specific characteristics. We're looking for a function with rational real coefficients, a leading coefficient of 3, and the roots $\sqrt{5}$ and 2. Let's break this down step-by-step, it's easier than you might think. This is a classic algebra problem, and understanding it will give you a solid foundation for tackling more complex math challenges. So, grab your pencils, and let's get started!

Understanding the Basics: Polynomials and Their Roots

Alright guys, before we jump into the solution, let's make sure we're all on the same page. A polynomial function is just a function that involves only non-negative integer powers of a variable (usually 'x') and coefficients (the numbers in front of the 'x's). The degree of a polynomial is the highest power of 'x' in the function. The roots of a polynomial are the values of 'x' that make the function equal to zero – they are the points where the function crosses the x-axis. A leading coefficient is the coefficient of the term with the highest power of x. For example, in the polynomial 3x^2 + 2x -1, the leading coefficient is 3. We can see that the question is asking us to construct a polynomial by knowing its roots. This is a very common task in mathematics, and it's essential to understand the connection between roots and polynomials. Furthermore, the constraint of having rational real coefficients is super important. It means all the numbers in our polynomial must be rational numbers (numbers that can be expressed as a fraction of two integers), and they must also be real numbers (no imaginary numbers allowed!). This constraint will affect how we determine the roots of the polynomial.

The Conjugate Root Theorem

Now, here's a crucial concept that helps us with this problem: the Conjugate Root Theorem. This theorem states that if a polynomial has rational coefficients, and if a complex number (or an irrational number involving a square root) is a root, then its conjugate is also a root. Because our polynomial needs to have rational real coefficients, and because we're given the root $\sqrt{5}$, this means that its conjugate, $-\sqrt{5}$, must also be a root. This is a huge clue! So, our polynomial has roots $2$, $\sqrt{5}$, and $-\sqrt{5}$. This means that the degree must be at least 3, because it can have a maximum number of roots equivalent to its degree. So we already have all the ingredients we need to solve the problem. Remember, whenever you see an irrational root like $\sqrt{5}$ in a polynomial with rational coefficients, automatically consider its conjugate.

Constructing the Polynomial Function

Okay, let's roll up our sleeves and build this polynomial, guys! We've established that our roots are $2$, $\sqrt{5}$, and $-\sqrt{5}$. We also know the leading coefficient should be 3. The general form of a polynomial with roots r1, r2, and r3 is:

f(x) = a * (x - r1) * (x - r2) * (x - r3)

where 'a' is the leading coefficient. In our case, a = 3, r1 = 2, r2 = $\sqrt{5}$, and r3 = $-\sqrt{5}$. Plugging these values in, we get:

f(x) = 3 * (x - 2) * (x - $\sqrt{5}$) * (x + $\sqrt{5}$)

Now we can start simplifying. Let's first multiply the terms involving the square roots:

(x - $\sqrt{5}$) * (x + $\sqrt{5}$) = x^2 - ($\sqrt{5}$)^2 = x^2 - 5

Great! So, we can rewrite our function as:

f(x) = 3 * (x - 2) * (x^2 - 5)

Now, let's multiply out the remaining terms:

f(x) = 3 * (x * (x^2 - 5) - 2 * (x^2 - 5)) f(x) = 3 * (x^3 - 5x - 2x^2 + 10) f(x) = 3 * (x^3 - 2x^2 - 5x + 10) f(x) = 3x^3 - 6x^2 - 15x + 30

And there we have it! Our polynomial function is f(x) = 3x^3 - 6x^2 - 15x + 30. This polynomial has a degree of 3, a leading coefficient of 3, rational real coefficients, and the roots 2, $\sqrt{5}$, and $-\sqrt{5}$.

Checking the Answer Choices

Now, let's compare our result with the answer choices you provided:

A. f(x) = 3x^3 - 6x^2 - 15x + 30 This matches our solution! B. f(x) = x^3 - 2x^2 - 5x + 10 This has the correct roots but doesn't have the required leading coefficient. C. f(x) = 3x^2 - 21x + 30 This is a quadratic, not a cubic and therefore, has only two roots. D. f(x) = x^2 - 7x + 10 This is also a quadratic and only has two roots.

So, by carefully working through the problem, considering the conjugate root theorem, and systematically constructing the polynomial, we arrived at the correct answer. The key here was to consider the conjugate of the irrational root and to ensure that the leading coefficient was correct. Always double-check your work and compare your final result with the options given to make sure you didn't make any errors along the way!

Conclusion: The Power of Polynomials

Well done, everyone! We've successfully navigated the world of polynomials and found the function that fits the criteria. This type of problem is super common in algebra and understanding the concepts of roots, coefficients, degree, and the conjugate root theorem is absolutely fundamental. We've shown that constructing a polynomial from its roots is a process that involves understanding the underlying mathematical principles and applying them step-by-step. Remember that the conjugate root theorem is your friend when dealing with polynomials with irrational or complex roots and rational coefficients. And always remember to double-check to make sure your answer makes sense based on the initial conditions.

Keep practicing these types of problems, and you'll become a polynomial pro in no time! Keep exploring the world of mathematics, and enjoy the journey! You've got this! Also, feel free to ask me if you have any questions, I'm happy to help you with that!