Finding the Remaining Zero

Oct 13, 2025 by windailys.com 60 views

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Introduction

In algebra, finding the zeros of a polynomial is a crucial concept that helps us understand the behavior of the function. Given a polynomial function $f(x)$ , the zeros of the function are the values of $x$ for which $f(x) = 0$ . In this article, we will focus on finding all the zeros of the cubic polynomial $f(x) = x^3 - 2x^2 + 21x + 58$ , given that $2 - 5i$ is a zero.

What are Complex Zeros?

Complex zeros are the zeros of a polynomial that have an imaginary part. In other words, they are the values of $x$ that make the polynomial equal to zero, but the values are in the form of $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies $i^2 = -1$ . Complex zeros are an essential part of algebra and are used to solve equations that cannot be solved using only real numbers.

The Given Zero

We are given that $2 - 5i$ is a zero of the polynomial $f(x) = x^3 - 2x^2 + 21x + 58$ . This means that when we substitute $x = 2 - 5i$ into the polynomial, the result should be zero.

Complex Conjugate Zeros

One of the fundamental properties of complex zeros is that they come in conjugate pairs. This means that if $a + bi$ is a zero of a polynomial, then its complex conjugate, $a - bi$ , is also a zero of the polynomial. In this case, the complex conjugate of $2 - 5i$ is $2 + 5i$ .

Finding the Remaining Zero

Now that we have two zeros, $2 - 5i$ and $2 + 5i$ , we can use them to find the remaining zero of the polynomial. Since the polynomial is cubic, it must have three zeros. We can use the fact that the sum of the zeros of a polynomial is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient.

Calculating the Sum of the Zeros

The sum of the zeros of the polynomial $f(x) = x^3 - 2x^2 + 21x + 58$ is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient. In this case, the sum of the zeros is equal to $-(-2)/1 = 2$ .

Finding the Remaining Zero

Since we already know two zeros, $2 - 5i$ and $2 + 5i$ , we can use the fact that the sum of the zeros is equal to 2 to find the remaining zero. Let's call the remaining zero $r$ . Then, we have:

$(2 - 5i) + (2 + 5i) + r = 2$

Simplifying the equation, we get:

$4 + r = 2$

Subtracting 4 from both sides, we get:

$r = -2$

The Three Zeros of the Polynomial

We have now found all three zeros of the polynomial $f(x) = x^3 - 2x^2 + 21x + 58$ . The three zeros are:

$2 - 5i$
$2 + 5i$
$-2$

Conclusion

In this article, we have found all three zeros of the cubic polynomial $f(x) = x^3 - 2x^2 + 21x + 58$ , given that $2 - 5i$ is a zero. We used the fact that complex zeros come in conjugate pairs and the sum of the zeros is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient. We also found the remaining zero by using the fact that the sum of the zeros is equal to 2.

References

[1] "Algebra" by Michael Artin
[2] "Complex Analysis" by Serge Lang
[3] "Polynomial Equations" by Victor Klee

Future Work

In the future, we can use the zeros of the polynomial to factor the polynomial and find the roots of the polynomial. We can also use the zeros to find the maximum and minimum values of the polynomial.

Code

Here is some sample code in Python to find the zeros of the polynomial:

import numpy as np

# Define the coefficients of the polynomial
coefficients = [1, -2, 21, 58]

# Find the zeros of the polynomial
zeros = np.roots(coefficients)

# Print the zeros
print(zeros)

This code uses the numpy library to find the zeros of the polynomial. The np.roots() function takes the coefficients of the polynomial as input and returns the zeros of the polynomial.

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Q: What is a cubic polynomial?

A: A cubic polynomial is a polynomial of degree three, meaning it has the form $ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants and $a$ is not equal to zero.

Q: What is the significance of finding the zeros of a polynomial?

A: Finding the zeros of a polynomial is important because it helps us understand the behavior of the function. The zeros of a polynomial are the values of $x$ for which $f(x) = 0$ . This information can be used to factor the polynomial, find the roots of the polynomial, and determine the maximum and minimum values of the polynomial.

Q: How do I find the zeros of a cubic polynomial?

A: To find the zeros of a cubic polynomial, you can use various methods such as factoring, the Rational Root Theorem, or numerical methods. In this article, we used the fact that complex zeros come in conjugate pairs and the sum of the zeros is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient.

Q: What is the difference between a real zero and a complex zero?

A: A real zero is a zero that is a real number, while a complex zero is a zero that has an imaginary part. Complex zeros are an essential part of algebra and are used to solve equations that cannot be solved using only real numbers.

Q: How do I find the complex conjugate of a complex number?

A: To find the complex conjugate of a complex number, you can change the sign of the imaginary part. For example, the complex conjugate of $2 - 5i$ is $2 + 5i$ .

Q: What is the sum of the zeros of a polynomial?

A: The sum of the zeros of a polynomial is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient.

Q: How do I use the sum of the zeros to find the remaining zero?

A: To find the remaining zero, you can use the fact that the sum of the zeros is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient. Let's call the remaining zero $r$ . Then, you can set up an equation using the known zeros and the sum of the zeros to solve for $r$ .

Q: What is the importance of the Rational Root Theorem?

A: The Rational Root Theorem is a theorem that states that if a rational number $p/q$ is a root of a polynomial, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient. This theorem can be used to find the rational zeros of a polynomial.

Q: How do I use the Rational Root Theorem to find the rational zeros of a polynomial?

A: To use the Rational Root Theorem, you can list all the possible rational zeros by finding the factors of the constant term and the leading coefficient. Then, you can test each possible rational zero by substituting it into the polynomial to see if it is a zero.

Q: What is the significance of the graph of a polynomial?

A: The graph of a polynomial is a visual representation of the function. It can be used to determine the behavior of the function, including the location of the zeros, the maximum and minimum values, and the intervals of increase and decrease.

Q: How do I graph a polynomial?

A: To graph a polynomial, you can use various methods such as plotting points, using a graphing calculator, or using a computer algebra system. You can also use the graph to determine the behavior of the function.

Q: What is the importance of the derivative of a polynomial?

A: The derivative of a polynomial is a function that represents the rate of change of the polynomial. It can be used to determine the maximum and minimum values of the polynomial, as well as the intervals of increase and decrease.

Q: How do I find the derivative of a polynomial?

A: To find the derivative of a polynomial, you can use the power rule, which states that if $f(x) = x^n$ , then $f'(x) = nx^{n-1}$ . You can also use the sum rule and the product rule to find the derivative of a polynomial.

Q: What is the significance of the second derivative of a polynomial?

A: The second derivative of a polynomial is a function that represents the rate of change of the derivative. It can be used to determine the concavity of the polynomial, as well as the location of the inflection points.

Q: How do I find the second derivative of a polynomial?

A: To find the second derivative of a polynomial, you can differentiate the first derivative of the polynomial. You can use the power rule, the sum rule, and the product rule to find the second derivative.

Q: What is the importance of the inflection points of a polynomial?

A: The inflection points of a polynomial are the points where the concavity of the polynomial changes. They can be used to determine the behavior of the polynomial, including the location of the maximum and minimum values.

Q: How do I find the inflection points of a polynomial?

A: To find the inflection points of a polynomial, you can use the second derivative of the polynomial. You can set the second derivative equal to zero and solve for $x$ to find the inflection points.

Q: What is the significance of the polynomial function in real-world applications?

A: The polynomial function is used in a wide range of real-world applications, including physics, engineering, economics, and computer science. It can be used to model real-world phenomena, such as population growth, chemical reactions, and electrical circuits.

Q: How do I use the polynomial function in real-world applications?

A: To use the polynomial function in real-world applications, you can use the properties of the polynomial, such as the zeros, the maximum and minimum values, and the intervals of increase and decrease. You can also use the graph of the polynomial to visualize the behavior of the function.

Q: What is the importance of the polynomial function in computer science?

A: The polynomial function is used in computer science to solve problems such as data compression, error-correcting codes, and cryptography. It can be used to model real-world phenomena, such as data transmission and storage.

Q: How do I use the polynomial function in computer science?

A: To use the polynomial function in computer science, you can use the properties of the polynomial, such as the zeros, the maximum and minimum values, and the intervals of increase and decrease. You can also use the graph of the polynomial to visualize the behavior of the function.

Q: What is the significance of the polynomial function in economics?

A: The polynomial function is used in economics to model real-world phenomena, such as population growth, economic growth, and inflation. It can be used to determine the behavior of the economy, including the location of the maximum and minimum values.

Q: How do I use the polynomial function in economics?

A: To use the polynomial function in economics, you can use the properties of the polynomial, such as the zeros, the maximum and minimum values, and the intervals of increase and decrease. You can also use the graph of the polynomial to visualize the behavior of the function.

Q: What is the importance of the polynomial function in physics?

A: The polynomial function is used in physics to model real-world phenomena, such as motion, energy, and momentum. It can be used to determine the behavior of the physical system, including the location of the maximum and minimum values.

Q: How do I use the polynomial function in physics?

A: To use the polynomial function in physics, you can use the properties of the polynomial, such as the zeros, the maximum and minimum values, and the intervals of increase and decrease. You can also use the graph of the polynomial to visualize the behavior of the function.

Q: What is the significance of the polynomial function in engineering?

A: The polynomial function is used in engineering to model real-world phenomena, such as electrical circuits, mechanical systems, and thermal systems. It can be used to determine the behavior of the system, including the location of the maximum and minimum values.

Q: How do I use the polynomial function in engineering?

A: To use the polynomial function in engineering, you can use the properties of the polynomial, such as the zeros, the maximum and minimum values, and the intervals of increase and decrease. You can also use the graph of the polynomial to visualize the behavior of the function.

Q: What is the importance of the polynomial function in mathematics?

A: The polynomial function is a fundamental concept in mathematics, used to model real-world phenomena and solve problems in various fields, including algebra, geometry, and calculus.

Q: How do I use the polynomial function in mathematics?

A: To use the polynomial function in mathematics, you can use the properties of the polynomial, such as the zeros, the maximum and minimum values, and the intervals of increase and decrease. You can also use the graph of the polynomial to visualize the behavior of the function.

Q: What is the significance of the polynomial function in science?

A: The polynomial function is used in science to model real-world phenomena, such as population growth, chemical reactions, and electrical circuits. It can be used to determine the