What Are The Zeros Of The Polynomial Function F ( X ) = X 3 − X 2 − 4 X + 4 F(x)=x^3-x^2-4x+4 F ( X ) = X 3 − X 2 − 4 X + 4 ?Select Each Correct Answer.-3 -2 -1 0 1 2 3

Mar 13, 2025 by ADMIN 175 views

**Finding the Zeros of a Polynomial Function: A Step-by-Step Guide**

===========================================================

Introduction

In mathematics, a polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a variable and a coefficient. The zeros of a polynomial function are the values of the variable that make the function equal to zero. In this article, we will focus on finding the zeros of the polynomial function $f(x)=x^3-x^2-4x+4$ .

Understanding Polynomial Functions

A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a variable and a coefficient. The general form of a polynomial function is:

f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0

where $a_n$ , $a_{n-1}$ , $\cdots$ , $a_1$ , and $a_0$ are constants, and $n$ is a non-negative integer.

The Rational Root Theorem

The rational root theorem is a theorem that states that if a rational number $p/q$ is a root of the polynomial function $f(x)$ , then $p$ must be a factor of the constant term $a_0$ , and $q$ must be a factor of the leading coefficient $a_n$ .

Applying the Rational Root Theorem to the Given Polynomial Function

To find the zeros of the polynomial function $f(x)=x^3-x^2-4x+4$ , we can apply the rational root theorem. The constant term $a_0$ is 4, and the leading coefficient $a_n$ is 1. Therefore, the possible rational roots of the polynomial function are:

\pm 1, \pm 2, \pm 4

Testing the Possible Rational Roots

We can test each of the possible rational roots by substituting them into the polynomial function and checking if the result is equal to zero.

Testing x = -3

Substituting x = -3 into the polynomial function, we get:

f(-3) = (-3)^3 - (-3)^2 - 4(-3) + 4

f(-3) = -27 - 9 + 12 + 4

f(-3) = -20

Since $f(-3)$ is not equal to zero, x = -3 is not a root of the polynomial function.

Testing x = -2

Substituting x = -2 into the polynomial function, we get:

f(-2) = (-2)^3 - (-2)^2 - 4(-2) + 4

f(-2) = -8 - 4 + 8 + 4

f(-2) = 0

Since $f(-2)$ is equal to zero, x = -2 is a root of the polynomial function.

Testing x = -1

Substituting x = -1 into the polynomial function, we get:

f(-1) = (-1)^3 - (-1)^2 - 4(-1) + 4

f(-1) = -1 - 1 + 4 + 4

f(-1) = 6

Since $f(-1)$ is not equal to zero, x = -1 is not a root of the polynomial function.

Testing x = 0

Substituting x = 0 into the polynomial function, we get:

f(0) = (0)^3 - (0)^2 - 4(0) + 4

f(0) = 4

Since $f(0)$ is not equal to zero, x = 0 is not a root of the polynomial function.

Testing x = 1

Substituting x = 1 into the polynomial function, we get:

f(1) = (1)^3 - (1)^2 - 4(1) + 4

f(1) = 1 - 1 - 4 + 4

f(1) = 0

Since $f(1)$ is equal to zero, x = 1 is a root of the polynomial function.

Testing x = 2

Substituting x = 2 into the polynomial function, we get:

f(2) = (2)^3 - (2)^2 - 4(2) + 4

f(2) = 8 - 4 - 8 + 4

f(2) = 0

Since $f(2)$ is equal to zero, x = 2 is a root of the polynomial function.

Testing x = 3

Substituting x = 3 into the polynomial function, we get:

f(3) = (3)^3 - (3)^2 - 4(3) + 4

f(3) = 27 - 9 - 12 + 4

f(3) = 10

Since $f(3)$ is not equal to zero, x = 3 is not a root of the polynomial function.

Conclusion

In conclusion, the zeros of the polynomial function $f(x)=x^3-x^2-4x+4$ are x = -2, x = 1, and x = 2.

Final Answer

The final answer is:

-2
1
2

=====================================================================================

Q: What is a polynomial function?

A: A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a variable and a coefficient.

Q: What is 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 the polynomial function $f(x)$ , then $p$ must be a factor of the constant term $a_0$ , and $q$ must be a factor of the leading coefficient $a_n$ .

Q: How do I apply the rational root theorem to find the zeros of a polynomial function?

A: To apply the rational root theorem, you need to find the factors of the constant term $a_0$ and the leading coefficient $a_n$ . Then, you can use these factors to test possible rational roots of the polynomial function.

Q: What are the possible rational roots of a polynomial function?

A: The possible rational roots of a polynomial function are the ratios of the factors of the constant term $a_0$ and the leading coefficient $a_n$ .

Q: How do I test the possible rational roots of a polynomial function?

A: To test the possible rational roots, you need to substitute each possible root into the polynomial function and check if the result is equal to zero.

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

A: Finding the zeros of a polynomial function is important because it helps us understand the behavior of the function and its graph. The zeros of a polynomial function are the points where the function intersects the x-axis.

Q: Can a polynomial function have more than one zero?

A: Yes, a polynomial function can have more than one zero. In fact, a polynomial function can have any number of zeros, including zero, one, or an infinite number of zeros.

Q: How do I find the zeros of a polynomial function with a degree greater than 3?

A: To find the zeros of a polynomial function with a degree greater than 3, you can use various methods such as the rational root theorem, synthetic division, or numerical methods.

Q: What are some common mistakes to avoid when finding the zeros of a polynomial function?

A: Some common mistakes to avoid when finding the zeros of a polynomial function include:

Not checking if the possible rational roots are actually roots of the polynomial function
Not using the correct method to test the possible rational roots
Not considering the possibility of complex roots
Not checking if the polynomial function has any repeated roots

Q: How do I check if a polynomial function has any repeated roots?

A: To check if a polynomial function has any repeated roots, you can use the fact that if a polynomial function has a repeated root, then the derivative of the polynomial function will also have that root.

Q: What are some real-world applications of finding the zeros of a polynomial function?

A: Some real-world applications of finding the zeros of a polynomial function include:

Modeling population growth and decline
Analyzing the behavior of electrical circuits
Studying the motion of objects under the influence of gravity
Optimizing the design of mechanical systems

Q: Can I use technology to find the zeros of a polynomial function?

A: Yes, you can use technology such as graphing calculators or computer software to find the zeros of a polynomial function. These tools can help you visualize the graph of the function and find the zeros more easily.

Q: How do I choose the right method to find the zeros of a polynomial function?

A: To choose the right method to find the zeros of a polynomial function, you need to consider the degree of the polynomial function, the complexity of the function, and the level of accuracy required.