Finding Remaining Zeros Of A Polynomial Of Degree 4

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Hey guys! Let's dive into the fascinating world of polynomials and their zeros. Today, we're tackling a problem where we need to find the remaining zeros of a degree 4 polynomial with rational coefficients, given some of its zeros. It sounds like a puzzle, right? Well, let's put on our detective hats and solve it together!

Understanding the Problem

So, the prompt gives us that we have a polynomial of degree 4. This is a key piece of information because it tells us that the polynomial has exactly four complex zeros (counting multiplicity), thanks to the Fundamental Theorem of Algebra. We're also told that the polynomial has rational coefficients. This is another important detail, as it brings into play the Conjugate Root Theorem. The Conjugate Root Theorem will be very useful in solving this problem as it gives us a way of finding other roots of the polynomial when we are given roots that are irrational and/or complex.

We're given three zeros: -1, √2, and 10/3. Our mission, should we choose to accept it (and we do!), is to find the remaining zero(s). Buckle up, because we're about to embark on a mathematical adventure!

The Conjugate Root Theorem: Our Secret Weapon

The Conjugate Root Theorem is our best friend in this scenario. It states that if a polynomial with rational coefficients has an irrational zero of the form a + √b, where a and b are rational and √b is irrational, then its conjugate, a - √b, is also a zero. Similarly, if a polynomial with real coefficients has a complex zero of the form a + bi, where a and b are real and i is the imaginary unit, then its complex conjugate, a - bi, is also a zero. This theorem is essential because it allows us to find other roots of a polynomial when we know that the coefficients of the polynomial are rational and we are given an irrational root.

In our case, we have √2 as a zero. We can rewrite √2 as 0 + √2. Since our polynomial has rational coefficients, the Conjugate Root Theorem tells us that 0 - √2, which is simply -√2, must also be a zero. This is a crucial step forward!

Putting the Pieces Together

So far, we know the following zeros: -1, √2, 10/3, and -√2. That's four zeros in total! Remember, our polynomial is of degree 4, which means it has exactly four zeros. We've found them all! Hooray!

Therefore, the remaining zero is -√2. We've successfully navigated the polynomial puzzle!

Steps to Find the Remaining Zero(s)

Let's recap the steps we took to solve this problem. This will help solidify our understanding and allow us to tackle similar problems in the future:

  1. Identify the given zeros: We started by listing the zeros provided in the problem: -1, √2, and 10/3.
  2. Apply the Conjugate Root Theorem: Recognizing that our polynomial has rational coefficients and √2 is a zero, we used the Conjugate Root Theorem to determine that -√2 is also a zero.
  3. Count the zeros: We counted the total number of zeros we had found: -1, √2, 10/3, and -√2. This gave us a total of four zeros.
  4. Consider the degree of the polynomial: We recalled that the polynomial is of degree 4, meaning it has exactly four zeros. Since we had found four zeros, we knew we had found them all.
  5. State the remaining zero(s): We concluded that the remaining zero is -√2.

A Detailed Example

Let's solidify our understanding with a more detailed walkthrough. Imagine we have a polynomial, P(x), of degree 4 with rational coefficients. We are given the zeros -1, √2, and 10/3. We need to find the other zero(s).

  1. Given zeros:

    • x = -1
    • x = √2
    • x = 10/3

  2. Applying the Conjugate Root Theorem:

    • Since √2 is a zero and the polynomial has rational coefficients, -√2 must also be a zero.

  3. List of all zeros found:

    • x = -1
    • x = √2
    • x = 10/3
    • x = -√2

  4. Degree of the polynomial:

    • The polynomial is of degree 4, meaning it has 4 zeros.

  5. Conclusion:

    • We have found all four zeros. The remaining zero is -√2.

Importance of Rational Coefficients

The condition that the polynomial has rational coefficients is absolutely crucial for the Conjugate Root Theorem to apply. If the coefficients were not rational, the conjugate of an irrational root wouldn't necessarily be a root itself. For example, consider a polynomial with irrational coefficients like (x - √2)(x - √3). This polynomial has √2 and √3 as roots, but their conjugates, -√2 and -√3, are not roots. This clearly shows that the rationality of the coefficients is a necessary condition for the Conjugate Root Theorem to hold. Make sure to always double-check this condition before applying the theorem!

Common Mistakes to Avoid

To make sure we're on the right track, let's discuss some common mistakes people make when tackling problems like this:

  • Forgetting the Conjugate Root Theorem: This is a big one! If you don't remember the Conjugate Root Theorem, you'll be stuck. Always keep it in mind when dealing with polynomials with rational coefficients and irrational or complex roots.
  • Not checking for rational coefficients: As we discussed, the Conjugate Root Theorem only applies to polynomials with rational coefficients. Make sure to verify this condition before using the theorem.
  • Incorrectly applying the Conjugate Root Theorem: Be careful to take the correct conjugate. For a root of the form a + √b, the conjugate is a - √b. For a complex root a + bi, the conjugate is a - bi. Pay close attention to the signs!
  • Not considering the degree of the polynomial: The degree of the polynomial tells you how many roots to expect. If you find fewer roots than the degree, you know you need to keep searching. If you find more, something went wrong!
  • Algebraic errors: Simple algebraic mistakes can throw off your entire solution. Double-check your work, especially when dealing with square roots and negative signs.

Practice Problems

Okay, now it's your turn to shine! Let's test your understanding with a few practice problems:

  1. A polynomial of degree 4 with rational coefficients has zeros 2, -√3, and 1 + i. Find the remaining zeros.
  2. A polynomial of degree 5 with rational coefficients has zeros -1, √5, and 2 - i. Find the remaining zeros.
  3. A polynomial of degree 3 with rational coefficients has a zero of 1 + √2. Find the other zeros.

Work through these problems, applying the steps and concepts we've discussed. Don't be afraid to make mistakes – that's how we learn! And remember, practice makes perfect.

Conclusion

Great job, guys! You've successfully learned how to find the remaining zeros of a polynomial with rational coefficients, given some of its zeros. We've explored the Conjugate Root Theorem, discussed common mistakes, and even tackled some practice problems. You're now well-equipped to handle similar challenges in the future. Remember, the key is to understand the underlying principles and apply them systematically. Keep practicing, keep exploring, and most importantly, keep enjoying the beauty and power of mathematics! This is just one piece of the puzzle of polynomial functions, there are lots more problems like these that involve different theorems that can be used to solve them. Remember to always take note of what the problem gives, and then use the information to see what tools you have available to you to solve the problem.