Determine If The Value 3 Is An Upper Bound For The Zeros Of The Function Shown Below.${ F(x) = -3x^3 + 20x^2 - 36x + 16 }$A. True B. False

Introduction

In mathematics, particularly in algebra and calculus, understanding the behavior of functions is crucial for solving various problems. One such concept is determining the upper bounds for the zeros of a function. This article will guide you through the process of finding upper bounds for the zeros of a given function and apply it to the function $f(x) = -3x^3 + 20x^2 - 36x + 16$ to determine if the value 3 is an upper bound for its zeros.

Understanding Upper Bounds

An upper bound for the zeros of a function is a value that is greater than or equal to all the zeros of the function. In other words, if we have a function $f(x)$ and an upper bound $M$, then all the zeros of $f(x)$ are less than or equal to $M$. Upper bounds are useful in various applications, such as finding the maximum or minimum values of a function, determining the number of zeros of a function, and solving equations.

Finding Upper Bounds

To find an upper bound for the zeros of a function, we can use various methods, including:

• Graphical method: Plotting the graph of the function and observing the behavior of the function in the region of interest.
• Numerical method: Using numerical methods, such as the bisection method or the secant method, to approximate the zeros of the function.
• Analytical method: Using algebraic manipulations, such as factoring or using the rational root theorem, to find the zeros of the function.

Applying the Rational Root Theorem

The rational root theorem states that if a rational number $p/q$ is a root of the polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, 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 $a_n$. We can use this theorem to find possible rational roots of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$.

Possible Rational Roots

Using the rational root theorem, we can find possible rational roots of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$. The factors of the constant term $16$ are $\pm 1, \pm 2, \pm 4, \pm 8, \pm 16$, and the factors of the leading coefficient $-3$ are $\pm 1, \pm 3$. Therefore, the possible rational roots of the function are $\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 1/3, \pm 2/3, \pm 4/3, \pm 8/3, \pm 16/3$.

Testing Possible Rational Roots

We can test these possible rational roots by substituting them into the function $f(x) = -3x^3 + 20x^2 - 36x + 16$ and checking if the result is zero. If the result is zero, then the corresponding rational root is a root of the function.

Determining the Upper Bound

After testing the possible rational roots, we find that the function $f(x) = -3x^3 + 20x^2 - 36x + 16$ has a root at $x = 1$. Since the function is a cubic function, it has at most three zeros. Therefore, the upper bound for the zeros of the function is the largest possible value that is greater than or equal to all the zeros of the function.

Conclusion

In conclusion, we have determined that the value 3 is not an upper bound for the zeros of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$. The function has a root at $x = 1$, and the upper bound for the zeros of the function is the largest possible value that is greater than or equal to all the zeros of the function.

Final Answer

The final answer is: B. False

References

• [1] Rational Root Theorem: A theorem in algebra that states that if a rational number $p/q$ is a root of the polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, 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 $a_n$.
• [2] Graphical Method: A method for finding the zeros of a function by plotting the graph of the function and observing the behavior of the function in the region of interest.
• [3] Numerical Method: A method for finding the zeros of a function by using numerical methods, such as the bisection method or the secant method, to approximate the zeros of the function.
• [4] Analytical Method: A method for finding the zeros of a function by using algebraic manipulations, such as factoring or using the rational root theorem, to find the zeros of the function.
  Q&A: Determining Upper Bounds for Zeros of a Function =====================================================

Q: What is an upper bound for the zeros of a function?

A: An upper bound for the zeros of a function is a value that is greater than or equal to all the zeros of the function.

Q: How do I find an upper bound for the zeros of a function?

A: You can use various methods to find an upper bound for the zeros of a function, including:

• Graphical method: Plotting the graph of the function and observing the behavior of the function in the region of interest.
• Numerical method: Using numerical methods, such as the bisection method or the secant method, to approximate the zeros of the function.
• Analytical method: Using algebraic manipulations, such as factoring or using the rational root theorem, to find the zeros of the function.

Q: What is the rational root theorem?

A: The rational root theorem is a theorem in algebra that states that if a rational number $p/q$ is a root of the polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, 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 use the rational root theorem to find possible rational roots of a function?

A: To use the rational root theorem to find possible rational roots of a function, you need to find the factors of the constant term and the leading coefficient of the function. Then, you can use these factors to find the possible rational roots of the function.

Q: What are the possible rational roots of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$?

A: The possible rational roots of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$ are $\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 1/3, \pm 2/3, \pm 4/3, \pm 8/3, \pm 16/3$.

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

A: To test possible rational roots of a function, you need to substitute the possible rational roots into the function and check if the result is zero. If the result is zero, then the corresponding rational root is a root of the function.

Q: What is the upper bound for the zeros of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$?

A: The upper bound for the zeros of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$ is the largest possible value that is greater than or equal to all the zeros of the function.

Q: Is the value 3 an upper bound for the zeros of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$?

A: No, the value 3 is not an upper bound for the zeros of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$. The function has a root at $x = 1$, and the upper bound for the zeros of the function is the largest possible value that is greater than or equal to all the zeros of the function.

Q: What are some common mistakes to avoid when determining upper bounds for zeros of a function?

A: Some common mistakes to avoid when determining upper bounds for zeros of a function include:

• Not using the correct method: Using the wrong method to find the upper bound for the zeros of a function can lead to incorrect results.
• Not testing all possible rational roots: Failing to test all possible rational roots of a function can lead to missing some of the zeros of the function.
• Not checking the result: Failing to check the result of substituting a possible rational root into the function can lead to incorrect conclusions.

Q: How can I practice determining upper bounds for zeros of a function?

A: You can practice determining upper bounds for zeros of a function by:

• Solving problems: Solving problems that involve determining upper bounds for zeros of a function can help you practice and improve your skills.
• Using online resources: Using online resources, such as video tutorials or practice problems, can help you learn and practice determining upper bounds for zeros of a function.
• Seeking help: Seeking help from a teacher or tutor can help you understand and practice determining upper bounds for zeros of a function.