Which Statement Describes The Graph Of This Polynomial Function?$f(x)=x^5-6x^4+9x^3$A. The Graph Crosses The $x$-axis At $x=0$ And Touches The $x$-axis At $x=3$. B. The Graph Touches The $x$-axis

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Understanding Polynomial Functions

Polynomial functions are a fundamental concept in algebra, and they play a crucial role in mathematics and its applications. 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. In this article, we will focus on the graph of a specific polynomial function, f(x)=x5−6x4+9x3f(x)=x^5-6x^4+9x^3, and determine which statement describes its graph.

Graphing Polynomial Functions

To graph a polynomial function, we need to understand its behavior and characteristics. The graph of a polynomial function can be described in terms of its degree, leading coefficient, and roots. The degree of a polynomial function is the highest power of the variable in the function. In this case, the degree of the function f(x)=x5−6x4+9x3f(x)=x^5-6x^4+9x^3 is 5.

Leading Coefficient and Roots

The leading coefficient of a polynomial function is the coefficient of the term with the highest power of the variable. In this case, the leading coefficient is 1. The roots of a polynomial function are the values of the variable that make the function equal to zero. To find the roots of the function f(x)=x5−6x4+9x3f(x)=x^5-6x^4+9x^3, we need to solve the equation x5−6x4+9x3=0x^5-6x^4+9x^3=0.

Solving the Equation

To solve the equation x5−6x4+9x3=0x^5-6x^4+9x^3=0, we can factor out the greatest common factor, which is x3x^3. This gives us x3(x2−6x+9)=0x^3(x^2-6x+9)=0. We can further factor the quadratic expression x2−6x+9x^2-6x+9 as (x−3)2(x-3)^2. Therefore, the equation becomes x3(x−3)2=0x^3(x-3)^2=0.

Finding the Roots

To find the roots of the equation x3(x−3)2=0x^3(x-3)^2=0, we need to set each factor equal to zero and solve for xx. This gives us x3=0x^3=0 and (x−3)2=0(x-3)^2=0. Solving for xx, we get x=0x=0 and x=3x=3.

Analyzing the Graph

Now that we have found the roots of the equation, we can analyze the graph of the function. The graph of a polynomial function can be described in terms of its behavior between its roots. In this case, the graph of the function f(x)=x5−6x4+9x3f(x)=x^5-6x^4+9x^3 has a root at x=0x=0 and another root at x=3x=3. Between these roots, the graph of the function is either increasing or decreasing.

Determining the Behavior of the Graph

To determine the behavior of the graph between its roots, we need to examine the leading coefficient and the degree of the function. The leading coefficient is 1, which means that the graph of the function is increasing between its roots. The degree of the function is 5, which means that the graph of the function is concave up between its roots.

Conclusion

In conclusion, the graph of the polynomial function f(x)=x5−6x4+9x3f(x)=x^5-6x^4+9x^3 has a root at x=0x=0 and another root at x=3x=3. The graph of the function is increasing between its roots and is concave up. Therefore, the statement that describes the graph of this polynomial function is:

A. The graph crosses the xx-axis at x=0x=0 and touches the xx-axis at x=3x=3

This statement accurately describes the behavior of the graph of the function f(x)=x5−6x4+9x3f(x)=x^5-6x^4+9x^3.

Final Thoughts

In this article, we have analyzed the graph of a polynomial function and determined which statement describes its behavior. We have used the concepts of leading coefficient, roots, and degree to understand the behavior of the graph. We have also examined the behavior of the graph between its roots and determined that it is increasing and concave up. This analysis has provided a deeper understanding of the graph of the polynomial function and has helped us to determine which statement accurately describes its behavior.

References

  • [1] "Polynomial Functions" by Math Open Reference
  • [2] "Graphing Polynomial Functions" by Purplemath
  • [3] "Leading Coefficient and Roots" by Khan Academy

Keywords

  • Polynomial functions
  • Graphing polynomial functions
  • Leading coefficient
  • Roots
  • Degree
  • Concave up
  • Increasing

Category

  • Mathematics

Discussion

  • This article provides a detailed analysis of the graph of a polynomial function and determines which statement describes its behavior.
  • The article uses the concepts of leading coefficient, roots, and degree to understand the behavior of the graph.
  • The article examines the behavior of the graph between its roots and determines that it is increasing and concave up.
  • The article provides a deeper understanding of the graph of the polynomial function and helps to determine which statement accurately describes its behavior.

Understanding Polynomial Functions

Polynomial functions are a fundamental concept in algebra, and they play a crucial role in mathematics and its applications. 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. In this article, we will answer some frequently asked questions about polynomial functions.

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: What is the degree of a polynomial function?

A: The degree of a polynomial function is the highest power of the variable in the function.

Q: What is the leading coefficient of a polynomial function?

A: The leading coefficient of a polynomial function is the coefficient of the term with the highest power of the variable.

Q: How do you find the roots of a polynomial function?

A: To find the roots of a polynomial function, you need to solve the equation when the function is equal to zero.

Q: What is the difference between a root and a zero of a polynomial function?

A: A root of a polynomial function is a value of the variable that makes the function equal to zero, while a zero of a polynomial function is a value of the variable that makes the function equal to zero and is also a root of the function.

Q: How do you determine the behavior of the graph of a polynomial function?

A: To determine the behavior of the graph of a polynomial function, you need to examine the leading coefficient and the degree of the function.

Q: What is the significance of the leading coefficient and the degree of a polynomial function?

A: The leading coefficient and the degree of a polynomial function determine the behavior of the graph of the function. The leading coefficient determines whether the graph is increasing or decreasing, while the degree determines whether the graph is concave up or concave down.

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

A: Yes, a polynomial function can have more than one root.

Q: Can a polynomial function have a root that is not an integer?

A: Yes, a polynomial function can have a root that is not an integer.

Q: How do you graph a polynomial function?

A: To graph a polynomial function, you need to find the roots of the function and then use the leading coefficient and the degree of the function to determine the behavior of the graph.

Q: What is the difference between a polynomial function and a rational 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, while a rational function is a function that can be expressed as the ratio of two polynomials.

Conclusion

In conclusion, polynomial functions are a fundamental concept in algebra, and they play a crucial role in mathematics and its applications. We have answered some frequently asked questions about polynomial functions, including what a polynomial function is, how to find the roots of a polynomial function, and how to determine the behavior of the graph of a polynomial function.

Final Thoughts

In this article, we have provided a comprehensive overview of polynomial functions and answered some frequently asked questions about them. We have also provided examples and explanations to help illustrate the concepts. We hope that this article has been helpful in understanding polynomial functions and their applications.

References

  • [1] "Polynomial Functions" by Math Open Reference
  • [2] "Graphing Polynomial Functions" by Purplemath
  • [3] "Leading Coefficient and Roots" by Khan Academy

Keywords

  • Polynomial functions
  • Graphing polynomial functions
  • Leading coefficient
  • Roots
  • Degree
  • Concave up
  • Increasing

Category

  • Mathematics

Discussion

  • This article provides a comprehensive overview of polynomial functions and answers some frequently asked questions about them.
  • The article provides examples and explanations to help illustrate the concepts.
  • The article is helpful in understanding polynomial functions and their applications.
  • The article is a valuable resource for students and teachers of mathematics.