What Are The Zeros Of The Polynomial Function F ( X ) = X 3 − 5 X 2 − 6 X F(x) = X^3 - 5x^2 - 6x F ( X ) = X 3 − 5 X 2 − 6 X ?A. X = − 1 , X = 0 X = -1, X = 0 X = − 1 , X = 0 , And X = 6 X = 6 X = 6 B. X = − 6 , X = 0 X = -6, X = 0 X = − 6 , X = 0 , And X = 1 X = 1 X = 1 C. X = − 3 , X = 0 X = -3, X = 0 X = − 3 , X = 0 , And X = 2 X = 2 X = 2

Mar 9, 2025 by ADMIN 345 views

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

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Introduction

In mathematics, a polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. The zeros of a polynomial function are the values of the variable that make the function equal to zero. In this article, we will explore how to find the zeros of a polynomial function, using the given function $F(x) = x^3 - 5x^2 - 6x$ as an example.

Understanding the Polynomial Function

The given polynomial function is $F(x) = x^3 - 5x^2 - 6x$ . To find the zeros of this function, we need to understand the properties of polynomial functions. A polynomial function of degree $n$ is a function that can be written in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$ , where $a_n \neq 0$ and $a_i$ are constants.

Factoring the Polynomial Function

One way to find the zeros of a polynomial function is to factor the function. Factoring a polynomial function involves expressing the function as a product of simpler polynomials. In this case, we can factor the given function as follows:

F(x) = x^3 - 5x^2 - 6x = x(x^2 - 5x - 6)

Finding the Zeros of the Factored Function

Now that we have factored the polynomial function, we can find the zeros of the function by setting each factor equal to zero and solving for $x$ . The first factor is $x$ , which is equal to zero when $x = 0$ . The second factor is $x^2 - 5x - 6$ , which can be factored further as $(x - 6)(x + 1)$ . Setting each of these factors equal to zero and solving for $x$ , we get:

x - 6 = 0 \Rightarrow x = 6

x + 1 = 0 \Rightarrow x = -1

Conclusion

In conclusion, the zeros of the polynomial function $F(x) = x^3 - 5x^2 - 6x$ are $x = 0$ , $x = -1$ , and $x = 6$ . These values of $x$ make the function equal to zero, and they can be found by factoring the polynomial function and setting each factor equal to zero.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Factor the polynomial function $F(x) = x^3 - 5x^2 - 6x$ as $x(x^2 - 5x - 6)$ .
Set each factor equal to zero and solve for $x$ $x$ :
- $x = 0$
- $x^2 - 5x - 6 = 0$
- $(x - 6)(x + 1) = 0$
- $x - 6 = 0 \Rightarrow x = 6$
- $x + 1 = 0 \Rightarrow x = -1$
The zeros of the polynomial function are $x = 0$ , $x = -1$ , and $x = 6$ .

Common Mistakes to Avoid

When finding the zeros of a polynomial function, there are several common mistakes to avoid:

Not factoring the polynomial function: Failing to factor the polynomial function can make it difficult to find the zeros of the function.
Not setting each factor equal to zero: Failing to set each factor equal to zero and solve for $x$ can result in missing zeros.
Not checking for extraneous solutions: Failing to check for extraneous solutions can result in incorrect zeros.

Real-World Applications

The zeros of a polynomial function have several real-world applications:

Engineering: The zeros of a polynomial function can be used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
Physics: The zeros of a polynomial function can be used to model and analyze physical systems, such as the motion of objects and the behavior of electrical circuits.
Economics: The zeros of a polynomial function can be used to model and analyze economic systems, such as the behavior of supply and demand curves.

Conclusion

In conclusion, finding the zeros of a polynomial function is an important concept in mathematics and has several real-world applications. By factoring the polynomial function and setting each factor equal to zero, we can find the zeros of the function. Additionally, by avoiding common mistakes and checking for extraneous solutions, we can ensure that our solutions are accurate and reliable.

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Q: What is a polynomial function?

A: A polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.

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

A: To find the zeros of a polynomial function, you need to factor the function and set each factor equal to zero. Then, solve for the variable to find the zeros of the function.

Q: What is factoring a polynomial function?

A: Factoring a polynomial function involves expressing the function as a product of simpler polynomials. This can help you find the zeros of the function more easily.

Q: How do I factor a polynomial function?

A: There are several methods for factoring a polynomial function, including:

Greatest Common Factor (GCF): This involves finding the greatest common factor of the terms in the polynomial function and factoring it out.
Difference of Squares: This involves factoring a polynomial function that can be written in the form $a^2 - b^2$ .
Sum and Difference: This involves factoring a polynomial function that can be written in the form $a^2 + b^2$ or $a^2 - b^2$ .

Q: What are the 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 factoring the polynomial function: Failing to factor the polynomial function can make it difficult to find the zeros of the function.
Not setting each factor equal to zero: Failing to set each factor equal to zero and solve for $x$ can result in missing zeros.
Not checking for extraneous solutions: Failing to check for extraneous solutions can result in incorrect zeros.

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

A: The zeros of a polynomial function have several real-world applications, including:

Engineering: The zeros of a polynomial function can be used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
Physics: The zeros of a polynomial function can be used to model and analyze physical systems, such as the motion of objects and the behavior of electrical circuits.
Economics: The zeros of a polynomial function can be used to model and analyze economic systems, such as the behavior of supply and demand curves.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solutions back into the original equation and check if they are true. If they are not true, then the solution is extraneous and should be discarded.

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

A: Finding the zeros of a polynomial function is an important concept in mathematics and has several real-world applications. By understanding how to find the zeros of a polynomial function, you can solve a wide range of problems in engineering, physics, and economics.

Q: Can you provide an example of finding the zeros of a polynomial function?

A: Here is an example of finding the zeros of a polynomial function:

Suppose we have the polynomial function $F(x) = x^3 - 5x^2 - 6x$ . To find the zeros of this function, we need to factor the function and set each factor equal to zero. Then, solve for the variable to find the zeros of the function.

First, we can factor the function as follows:

F(x) = x^3 - 5x^2 - 6x = x(x^2 - 5x - 6)

Next, we can set each factor equal to zero and solve for $x$ :

x = 0

x^2 - 5x - 6 = 0

(x - 6)(x + 1) = 0

x - 6 = 0 \Rightarrow x = 6

x + 1 = 0 \Rightarrow x = -1

Therefore, the zeros of the polynomial function $F(x) = x^3 - 5x^2 - 6x$ are $x = 0$ , $x = -1$ , and $x = 6$ .

Q: Can you provide a summary of the key concepts in finding the zeros of a polynomial function?

A: Here is a summary of the key concepts in finding the zeros of a polynomial function:

Factoring the polynomial function: This involves expressing the function as a product of simpler polynomials.
Setting each factor equal to zero: This involves setting each factor equal to zero and solving for the variable to find the zeros of the function.
Checking for extraneous solutions: This involves plugging the solutions back into the original equation and checking if they are true.
Real-world applications: The zeros of a polynomial function have several real-world applications, including engineering, physics, and economics.

By understanding these key concepts, you can find the zeros of a polynomial function and solve a wide range of problems in mathematics and other fields.