Polynomial Function Lowest Degree With Rational Coefficients Explained

by Henrik Larsen 71 views

Hey guys! Ever find yourself staring at a math problem that looks like it’s written in another language? Polynomial functions can seem intimidating, but once you break them down, they’re actually pretty cool. Let's tackle a fun one together, focusing on finding the polynomial function of the lowest degree with rational real coefficients, a leading coefficient of 3, and roots of $\sqrt{5}$ and 2. Buckle up, because we’re about to dive deep into the world of polynomials!

Understanding the Basics: What Are We Looking For?

Before we jump into solving the problem, let’s make sure we’re all on the same page with some key concepts. Polynomial functions are expressions that involve variables raised to non-negative integer powers, combined with coefficients. For example, $3x^2 + 2x - 1$ is a polynomial function. The degree of a polynomial is the highest power of the variable (in this case, 2). Roots (also called zeros) are the values of x that make the function equal to zero. And, the leading coefficient is the number multiplied by the highest power of x.

So, in our problem, we need to find a polynomial that:

  1. Has rational real coefficients. This means the numbers in front of the x terms (and the constant term) should be rational numbers (fractions or integers).
  2. Has a leading coefficient of 3. This means the number in front of the highest power of x is 3.
  3. Has roots of $\sqrt{5}$ and 2. This means that if you plug in $\sqrt{5}$ or 2 for x, the polynomial will equal zero.
  4. Has the lowest degree possible. We want the simplest polynomial that satisfies the conditions.

Complex Conjugate Root Theorem

Now, here’s a crucial piece of the puzzle: the Complex Conjugate Root Theorem. This theorem states that if a polynomial with real coefficients has a complex root (a + bi, where i is the imaginary unit), then its complex conjugate (a - bi) must also be a root. A similar principle applies to irrational roots. If a polynomial with rational coefficients has an irrational root of the form a + $\sqrt{b}$, where a and b are rational and $\sqrt{b}$ is irrational, then its conjugate a - $\sqrt{b}$ must also be a root.

In our case, we have a root of $\sqrt5}$, which can be written as 0 + $\sqrt{5}$. Since the polynomial has rational coefficients, the conjugate 0 - $\sqrt{5}$, or -$\sqrt{5}$, must also be a root. This is a critical insight because it means our polynomial must have at least three roots $\sqrt{5$, -$\sqrt{5}$, and 2. This tells us that the polynomial will be at least a cubic (degree 3) polynomial.

Building the Polynomial: Putting the Pieces Together

Okay, now we know we need a cubic polynomial with roots $\sqrt{5}$, -$\sqrt{5}$, and 2. How do we build it? We can use the fact that if 'r' is a root of a polynomial, then (x - r) is a factor of the polynomial. So, our polynomial will have the following factors:

  • (x - $\sqrt{5}$)
  • (x + $\sqrt{5}$) (because x - (-$\sqrt{5}$) = x + $\sqrt{5}$)
  • (x - 2)

To get the polynomial, we multiply these factors together. First, let's multiply the factors involving the square roots:

(x−sqrt5)(x+sqrt5)=x2−(sqrt5)2=x2−5(x - \\sqrt{5})(x + \\sqrt{5}) = x^2 - (\\sqrt{5})^2 = x^2 - 5

This is a neat trick! Multiplying conjugates like this eliminates the square roots, giving us a polynomial with rational coefficients. Now, we multiply this result by the remaining factor (x - 2):

(x2−5)(x−2)=x3−2x2−5x+10(x^2 - 5)(x - 2) = x^3 - 2x^2 - 5x + 10

This gives us a polynomial with roots $\sqrt{5}$, -$\sqrt{5}$, and 2. However, we need a leading coefficient of 3. To achieve this, we simply multiply the entire polynomial by 3:

3(x3−2x2−5x+10)=3x3−6x2−15x+303(x^3 - 2x^2 - 5x + 10) = 3x^3 - 6x^2 - 15x + 30

Voilà! We’ve found our polynomial.

The Solution and Why It's the Lowest Degree

The polynomial function of the lowest degree with rational real coefficients, a leading coefficient of 3, and roots $\sqrt{5}$ and 2 is:

f(x)=3x3−6x2−15x+30f(x) = 3x^3 - 6x^2 - 15x + 30

So, the correct answer is A. But let's quickly recap why this is the lowest possible degree.

We started with two given roots, $\sqrt{5}$ and 2. Because the polynomial needs to have rational coefficients, the irrational root $\sqrt{5}$ forces its conjugate, -$\sqrt{5}$, to also be a root. This gives us a minimum of three roots. A polynomial has at least the same degree as the number of roots. Hence, we needed a cubic (degree 3) polynomial. We constructed a cubic polynomial that fits all the criteria, so we know it's the lowest possible degree.

Avoiding Common Pitfalls

When tackling problems like this, there are a few common mistakes to watch out for:

  • Forgetting the conjugate root: This is the biggest pitfall. If you’re given an irrational or complex root and the problem specifies rational (or real) coefficients, remember to include its conjugate as a root as well.
  • Incorrectly multiplying factors: Double-check your multiplication when expanding the factors. Simple arithmetic errors can throw off the entire solution.
  • Ignoring the leading coefficient: Don’t forget to adjust the polynomial to match the specified leading coefficient.
  • Assuming the given roots are the only roots: In some cases, the polynomial might have more roots. However, we were asked for the lowest degree polynomial, so we only considered the minimum number of roots required.

Practice Makes Perfect: How to Master Polynomial Problems

Polynomial problems might seem tricky at first, but with practice, you'll become a pro. Here are a few tips for mastering these types of questions:

  1. Review the fundamental theorems: Make sure you have a solid grasp of the Complex Conjugate Root Theorem (and its counterpart for irrational roots), the Factor Theorem, and the Remainder Theorem.
  2. Practice factoring polynomials: Being comfortable with factoring will help you work backwards from roots to polynomials.
  3. Work through examples: The more problems you solve, the better you’ll become at recognizing patterns and applying the correct techniques.
  4. Check your work: Always take a moment to review your calculations and make sure your answer makes sense in the context of the problem.

Real-World Applications: Where Polynomials Pop Up

You might be wondering, “Okay, this is cool, but where do polynomials actually matter in the real world?” Well, you’d be surprised! Polynomial functions are used extensively in various fields:

  • Engineering: Polynomials are used to model curves and shapes, design structures, and analyze systems.
  • Physics: They appear in equations describing projectile motion, electrical circuits, and quantum mechanics.
  • Computer Graphics: Polynomials are used to create smooth curves and surfaces in computer graphics and animations.
  • Economics: They can be used to model cost and revenue functions.
  • Statistics: Polynomial regression is a technique used to fit curves to data.

So, understanding polynomials isn't just about passing a math test; it’s about gaining a fundamental tool for understanding and modeling the world around us.

Conclusion: You've Got This!

Finding the polynomial function with specific characteristics might seem like a daunting task, but by breaking it down step by step, we can conquer even the trickiest problems. Remember the key concepts: roots, factors, conjugates, and leading coefficients. With a little practice, you'll be solving polynomial puzzles like a math whiz! Keep practicing, and don't be afraid to ask questions. You've got this!

Polynomial Function Lowest Degree with Rational Coefficients Explained

Find the polynomial function of the lowest degree with rational real coefficients, a leading coefficient of 3, and roots $\sqrt{5}$ and 2. Explain the steps.