What Are The Zeros Of The Polynomial Function?${ F(x) = X^3 - X^2 - 4x + 4 }$Select Each Correct Answer.A. { -3$}$B. { -2$}$C. { -1$}$D. 0E. 1F. 2G. 3

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 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 explore the concept of zeros of a polynomial function and find the zeros of the given polynomial function $f(x) = x^3 - x^2 - 4x + 4$.

What are Zeros of a Polynomial Function?

The zeros of a polynomial function are the values of the variable that make the function equal to zero. In other words, if we substitute a zero of the function into the function, the result will be zero. For example, if we have a function $f(x) = x^2 - 4$, the zeros of the function are $x = 2$ and $x = -2$, because when we substitute $x = 2$ or $x = -2$ into the function, the result is zero.

Finding Zeros of a Polynomial Function

There are several methods to find the zeros of a polynomial function, including factoring, the Rational Root Theorem, and the use of a graphing calculator. In this article, we will use the Rational Root Theorem to find the zeros of the given polynomial function.

The Rational Root Theorem

The Rational Root Theorem states that if a rational number $p/q$ is a zero of the polynomial function $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, where $p$ and $q$ are integers and $q$ is non-zero, then $p$ must be a factor of the constant term $a_0$ and $q$ must be a factor of the leading coefficient $a_n$.

Finding the Zeros of the Given Polynomial Function

To find the zeros of the given polynomial function $f(x) = x^3 - x^2 - 4x + 4$, we can use the Rational Root Theorem. The constant term of the function is 4, and the leading coefficient is 1. Therefore, the possible rational zeros of the function are the factors of 4, which are $\pm 1$, $\pm 2$, and $\pm 4$.

Testing the Possible Rational Zeros

We can test each of the possible rational zeros by substituting them into the function and checking if the result is zero. If the result is zero, then the value is a zero of the function.

• For $x = -4$, we have $f(-4) = (-4)^3 - (-4)^2 - 4(-4) + 4 = -64 - 16 + 16 + 4 = -60$, which is not zero.
• For $x = -2$, we have $f(-2) = (-2)^3 - (-2)^2 - 4(-2) + 4 = -8 - 4 + 8 + 4 = 0$, which is zero.
• For $x = -1$, we have $f(-1) = (-1)^3 - (-1)^2 - 4(-1) + 4 = -1 - 1 + 4 + 4 = 6$, which is not zero.
• For $x = 1$, we have $f(1) = (1)^3 - (1)^2 - 4(1) + 4 = 1 - 1 - 4 + 4 = 0$, which is zero.
• For $x = 2$, we have $f(2) = (2)^3 - (2)^2 - 4(2) + 4 = 8 - 4 - 8 + 4 = 0$, which is zero.
• For $x = 4$, we have $f(4) = (4)^3 - (4)^2 - 4(4) + 4 = 64 - 16 - 16 + 4 = 36$, which is not zero.

Conclusion

In this article, we have found the zeros of the given polynomial function $f(x) = x^3 - x^2 - 4x + 4$ using the Rational Root Theorem. The zeros of the function are $x = -2$, $x = 1$, and $x = 2$. These values make the function equal to zero, and therefore, they are the zeros of the function.

Final Answer

The final answer is:

• A. $\boxed{-2}$
• B. $\boxed{1}$
• C. $\boxed{2}$
  Q&A: Zeros of a Polynomial Function =====================================

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 zero of the polynomial function $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, where $p$ and $q$ are integers and $q$ is non-zero, 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 find the zeros of a polynomial function using the Rational Root Theorem?

A: To find the zeros of a polynomial function using the Rational Root Theorem, you need to follow these steps:

1. Identify the constant term and the leading coefficient of the polynomial function.
2. List all the factors of the constant term and the leading coefficient.
3. Create a list of possible rational zeros by dividing each factor of the constant term by each factor of the leading coefficient.
4. Test each possible rational zero by substituting it into the polynomial function and checking if the result is zero.
5. If the result is zero, then the value is a zero of the function.

Q: What are the possible rational zeros of the polynomial function $f(x) = x^3 - x^2 - 4x + 4$?

A: The possible rational zeros of the polynomial function $f(x) = x^3 - x^2 - 4x + 4$ are the factors of the constant term 4, which are $\pm 1$, $\pm 2$, and $\pm 4$.

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

A: To test the possible rational zeros of a polynomial function, you need to substitute each possible rational zero into the polynomial function and check if the result is zero. If the result is zero, then the value is a zero of the function.

Q: What are the zeros of the polynomial function $f(x) = x^3 - x^2 - 4x + 4$?

A: The zeros of the polynomial function $f(x) = x^3 - x^2 - 4x + 4$ are $x = -2$, $x = 1$, and $x = 2$. These values make the function equal to zero, and therefore, they are the zeros of the function.

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

A: Yes, you can use a graphing calculator to find the zeros of a polynomial function. A graphing calculator can graph the polynomial function and find the x-intercepts, which are the zeros of the function.

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 following the steps of the Rational Root Theorem correctly.
• Not testing all possible rational zeros.
• Not checking if the result is zero when testing a possible rational zero.
• Not using a graphing calculator to check the x-intercepts of the polynomial function.

Q: How do I determine if a value is a zero of a polynomial function?

A: To determine if a value is a zero of a polynomial function, you need to substitute the value into the polynomial function and check if the result is zero. If the result is zero, then the value is a zero of the function.

Q: Can I use the Rational Root Theorem to find the zeros of a polynomial function with a non-rational zero?

A: No, the Rational Root Theorem only applies to polynomial functions with rational zeros. If a polynomial function has a non-rational zero, then the Rational Root Theorem cannot be used to find the zero.

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 motion of objects.
• Finding the maximum or minimum value of a function.
• Solving systems of equations.

Q: How do I apply the Rational Root Theorem to find the zeros of a polynomial function in a real-world context?

A: To apply the Rational Root Theorem to find the zeros of a polynomial function in a real-world context, you need to:

1. Identify the problem and the polynomial function that models it.
2. Use the Rational Root Theorem to find the possible rational zeros of the polynomial function.
3. Test each possible rational zero to find the actual zeros of the function.
4. Use the zeros of the function to solve the problem.

Q: What are some common challenges when finding the zeros of a polynomial function?

A: Some common challenges when finding the zeros of a polynomial function include:

• Finding the correct factors of the constant term and the leading coefficient.
• Testing all possible rational zeros.
• Checking if the result is zero when testing a possible rational zero.
• Using a graphing calculator to check the x-intercepts of the polynomial function.

Q: How do I overcome these challenges when finding the zeros of a polynomial function?

A: To overcome these challenges when finding the zeros of a polynomial function, you need to:

1. Follow the steps of the Rational Root Theorem carefully.
2. Test all possible rational zeros.
3. Check if the result is zero when testing a possible rational zero.
4. Use a graphing calculator to check the x-intercepts of the polynomial function.
5. Practice finding the zeros of polynomial functions to become more confident and proficient.