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

Nov 10, 2025 by Admin 63 views

In this comprehensive guide, we'll dive into the fascinating world of polynomial functions, specifically focusing on a 5th-degree polynomial with rational coefficients. Our mission? To uncover all the hidden zeros, given a few clues: $-\frac{1}{6}$ , $\sqrt{11}$ , and $-4i$ . Buckle up, because we're about to embark on a mathematical adventure that's both informative and engaging!

Understanding the Fundamentals

Before we jump into solving the problem, let's establish a solid understanding of the key concepts involved. This will not only help you grasp the solution better but also equip you with the knowledge to tackle similar problems with confidence.

Polynomial Functions

A polynomial function is a mathematical expression consisting of variables and coefficients, combined using only addition, subtraction, and non-negative integer exponents. The general form of a polynomial function is:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$

where:

$a_n, a_{n-1}, ..., a_1, a_0$ are the coefficients (constants).
$x$ is the variable.
$n$ is a non-negative integer representing the degree of the polynomial.

In our case, we're dealing with a 5th-degree polynomial, meaning the highest power of $x$ is 5.

Zeros of a Polynomial

The zeros of a polynomial function are the values of $x$ that make the function equal to zero. In other words, they are the solutions to the equation $f(x) = 0$ . These zeros are also known as roots or x-intercepts of the polynomial function.

Rational Coefficients

A polynomial with rational coefficients means that all the coefficients ( $a_n, a_{n-1}, ..., a_1, a_0$ ) are rational numbers. Rational numbers are numbers that can be expressed as a fraction $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ .

The Conjugate Root Theorem

This theorem is crucial for solving our problem. It states that if a polynomial with real coefficients has a complex root $a + bi$ , then its complex conjugate $a - bi$ is also a root. Similarly, if a polynomial with rational coefficients has an irrational root $a + \sqrt{b}$ , where $a$ and $b$ are rational numbers and $\sqrt{b}$ is irrational, then its conjugate $a - \sqrt{b}$ is also a root. This is especially important for our problem. Knowing this theorem is essential in finding the missing zeros.

Applying the Conjugate Root Theorem

Now that we have a firm grasp of the fundamentals, let's apply the Conjugate Root Theorem to find the missing zeros of our 5th-degree polynomial.

We are given the following zeros:

$-\frac{1}{6}$ (rational)
$\sqrt{11}$ (irrational)
$-4i$ (complex)

Since the polynomial has rational coefficients, we can use the Conjugate Root Theorem to find the other zeros.

For the irrational root $\sqrt{11}$ , its conjugate $-\sqrt{11}$ must also be a root.
For the complex root $-4i$ , which can be written as $0 - 4i$ , its conjugate $0 + 4i$ , or simply $4i$ , must also be a root.

Therefore, the other two zeros are $-\sqrt{11}$ and $4i$ .

Finding All the Zeros

Now we know all five zeros of the 5th-degree polynomial:

$-\frac{1}{6}$
$\sqrt{11}$
$-\sqrt{11}$
$-4i$
$4i$

Since a 5th-degree polynomial has exactly five roots (counting multiplicity), we have found all the zeros. It's important to remember that a polynomial of degree n has n roots. This is a fundamental concept in algebra. Understanding this concept makes solving these problems much easier.

Constructing the Polynomial (Optional)

Although the problem only asks for the other zeros, let's go a step further and construct the polynomial function. This will solidify our understanding and demonstrate how the zeros relate to the polynomial.

If we know the zeros of a polynomial, we can write it in factored form:

$f(x) = a(x - r_1)(x - r_2)(x - r_3)(x - r_4)(x - r_5)$

where:

$a$ is a constant (leading coefficient).
$r_1, r_2, r_3, r_4, r_5$ are the zeros.

Plugging in our zeros, we get:

$f(x) = a(x + \frac{1}{6})(x - \sqrt{11})(x + \sqrt{11})(x + 4i)(x - 4i)$

To simplify, we can multiply the conjugate pairs:

$(x - \sqrt{11})(x + \sqrt{11}) = x^2 - 11$

$(x + 4i)(x - 4i) = x^2 + 16$

So, our polynomial becomes:

$f(x) = a(x + \frac{1}{6})(x^2 - 11)(x^2 + 16)$

To get rid of the fraction, we can multiply the factor $(x + \frac{1}{6})$ by 6:

$f(x) = a(6x + 1)(x^2 - 11)(x^2 + 16)$

Expanding this expression would give us the polynomial in standard form. Note that the value of 'a' can be any non-zero rational number, and it will simply scale the polynomial vertically. For simplicity, we can assume a=1.

Common Mistakes to Avoid

When solving problems like this, it's easy to make mistakes. Here are a few common pitfalls to watch out for:

Forgetting the Conjugate Root Theorem: This is the most common mistake. Always remember that irrational and complex roots come in conjugate pairs for polynomials with rational or real coefficients, respectively.
Incorrectly Identifying Conjugates: Make sure you correctly identify the conjugate of a complex or irrational number. The conjugate of $a + bi$ is $a - bi$ , and the conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$ .
Not Considering All Roots: A polynomial of degree $n$ has exactly $n$ roots (counting multiplicity). Make sure you find all the roots before concluding your solution.
Algebraic Errors: Be careful with your algebra, especially when expanding and simplifying expressions. Double-check your work to avoid mistakes.

Practice Problems

To solidify your understanding, try solving these practice problems:

A polynomial function of degree 4 with rational coefficients has zeros $2 - \sqrt{3}$ and $1 + i$ . Find the other zeros.
A polynomial function of degree 5 with real coefficients has zeros $3$ , $2i$ , and $-i$ . Find the other zeros.
A polynomial function of degree 6 with rational coefficients has zeros $\frac{1}{2}$ , $-\sqrt{5}$ , and $1 - 2i$ . Find the other zeros.

Conclusion

Finding the zeros of a polynomial function, especially when given some initial zeros and the degree of the polynomial, can seem daunting at first. However, by understanding the fundamental concepts, such as the definition of polynomial functions, zeros, rational coefficients, and the crucial Conjugate Root Theorem, you can confidently tackle these problems. Remember to practice and pay attention to common mistakes to avoid pitfalls. With this knowledge, you're well-equipped to unravel the mysteries of polynomial functions and their zeros! Understanding these concepts is essential for success in algebra and beyond.

So, to recap, given the zeros $-\frac{1}{6}, \sqrt{11}, -4i$ of a 5th-degree polynomial with rational coefficients, the other zeros are $-\sqrt{11}$ and $4i$ . Keep practicing, and you'll become a pro at finding polynomial zeros in no time! Good luck, guys!