Finding Zeros Of A 5th Degree Polynomial: A Step-by-Step Guide

by Alex Johnson 63 views

Let's dive into the fascinating world of polynomials! In this guide, we'll tackle the challenge of finding the zeros of a 5th degree polynomial, especially when given some initial zeros and the knowledge that the coefficients are rational. This type of problem often appears in mathematics courses, and understanding the underlying principles is key to solving it effectively. So, grab your thinking caps, and let’s get started!

Understanding the Problem

In this scenario, we're working with a 5th degree polynomial function. This means the highest power of the variable (usually x) is 5. Polynomials of this degree can be a bit tricky to handle directly, but luckily, we have some powerful theorems and properties to help us. Our specific challenge is that we know some of the zeros (also called roots) of the polynomial: 6, -4 + 5i, and 5 - √7. Our mission is to find the remaining zero(s). The fact that the polynomial has rational coefficients is a crucial piece of information, as it allows us to leverage the Complex Conjugate Root Theorem and the Irrational Conjugate Root Theorem.

Key Concepts to Remember:

  • Zeros/Roots: These are the values of x that make the polynomial equal to zero. Graphically, they're the points where the polynomial crosses the x-axis.
  • Degree of a Polynomial: The highest power of the variable in the polynomial. A 5th degree polynomial has five roots (counting multiplicity).
  • Rational Coefficients: This means all the numbers in front of the x terms (and the constant term) are rational numbers (can be expressed as a fraction p/q, where p and q are integers and q ≠ 0).
  • Complex Conjugate Root Theorem: If a polynomial with real coefficients has a complex number (a + bi) as a root, then its complex conjugate (a - bi) is also a root.
  • Irrational Conjugate Root Theorem: If a polynomial with rational coefficients has an irrational number (a + √b) as a root, then its conjugate (a - √b) is also a root.

Applying the Conjugate Root Theorems

This is where the magic happens! The Complex Conjugate Root Theorem and the Irrational Conjugate Root Theorem are our secret weapons in solving this problem. Because our polynomial has rational coefficients, we can use these theorems to deduce additional roots from the ones we already know.

  1. Dealing with the Complex Root: We are given that -4 + 5i is a zero. This is a complex number, with -4 as the real part and 5 as the imaginary part. The Complex Conjugate Root Theorem tells us that the complex conjugate, -4 - 5i, must also be a zero of the polynomial. So, we've just found another root!

  2. Addressing the Irrational Root: We also know that 5 - √7 is a zero. This is an irrational number because it involves the square root of 7, which is not a perfect square. The Irrational Conjugate Root Theorem states that the conjugate, 5 + √7, is also a zero. We've discovered yet another root!

By applying these theorems, we've expanded our list of zeros significantly. We started with three zeros and now have five: 6, -4 + 5i, -4 - 5i, 5 - √7, and 5 + √7.

Finding the Remaining Zero(s)

Now, let's piece it all together. Remember that we're dealing with a 5th degree polynomial. A fundamental theorem of algebra tells us that a polynomial of degree n has exactly n complex roots (counting multiplicity). Since our polynomial is of degree 5, it has five roots. We've already identified five zeros: 6, -4 + 5i, -4 - 5i, 5 - √7, and 5 + √7. This means we've found all the zeros of the polynomial!

Therefore, the other zeros are -4 - 5i and 5 + √7.

It’s worth noting that if we had been given a polynomial of a higher degree, say degree 6 or 7, and only three initial zeros, we would have used the conjugate root theorems to find additional roots and then potentially employed other techniques (like polynomial division or factoring) to find any remaining zeros.

Constructing the Polynomial (Optional)

While the question asks us to find the zeros, it's insightful to consider how we could construct the polynomial itself. This step isn’t strictly necessary to answer the original question, but it reinforces our understanding of the relationship between zeros and polynomials. To construct the polynomial, we can use the fact that if r is a zero of a polynomial, then (x - r) is a factor of the polynomial.

We have five zeros: 6, -4 + 5i, -4 - 5i, 5 - √7, and 5 + √7. This means our polynomial can be written in the form:

P(x) = a(x - 6)(x - (-4 + 5i))(x - (-4 - 5i))(x - (5 - √7))(x - (5 + √7))

where a is a constant. Let's break down the multiplication step by step:

  1. Multiply the complex conjugate factors:

(x - (-4 + 5i))(x - (-4 - 5i)) = (x + 4 - 5i)(x + 4 + 5i)

This looks intimidating, but we can use the difference of squares pattern: (A - B)(A + B) = A² - B². Here, A = (x + 4) and B = 5i.

So, we get:

(x + 4)² - (5i)² = x² + 8x + 16 - (-25) = x² + 8x + 41

  1. Multiply the irrational conjugate factors:

(x - (5 - √7))(x - (5 + √7)) = (x - 5 + √7)(x - 5 - √7)

Again, we use the difference of squares pattern: (A - B)(A + B) = A² - B². Here, A = (x - 5) and B = √7.

This gives us:

(x - 5)² - (√7)² = x² - 10x + 25 - 7 = x² - 10x + 18

  1. Now we have:

P(x) = a(x - 6)(x² + 8x + 41)(x² - 10x + 18)

To get the polynomial in standard form, we would need to multiply these factors together. This can be a bit tedious, but it's a straightforward process.

  1. Choosing a value for a: Since the problem only asks for the zeros, the value of a doesn't affect the roots. We can choose a = 1 for simplicity. If we were given another condition, like the value of the polynomial at a specific point, we could use that to determine the value of a.

By carrying out the multiplication (which we won't do in detail here), we would obtain the 5th degree polynomial in standard form.

Key Takeaways

Let's recap the essential points we've covered:

  • Conjugate Root Theorems: The Complex Conjugate Root Theorem and the Irrational Conjugate Root Theorem are powerful tools for finding zeros of polynomials with rational coefficients.
  • Degree and Number of Roots: A polynomial of degree n has exactly n complex roots (counting multiplicity).
  • Building the Polynomial: Knowing the zeros allows us to construct the polynomial by using the factors (x - r), where r is a zero.

Conclusion

Finding the zeros of a polynomial, especially a 5th degree polynomial, can seem daunting at first. However, by understanding and applying the Complex Conjugate Root Theorem and the Irrational Conjugate Root Theorem, we can systematically uncover the hidden roots. Remember to break down the problem into smaller steps, and don't be afraid to use the theorems to your advantage. With practice and a solid grasp of the fundamentals, you'll be solving these types of problems like a pro! For further reading on polynomial functions and the theorems discussed, check out trusted resources like Khan Academy's Polynomial Arithmetic section.