At Which Root Does The Graph Of F ( X ) = ( X − 5 ) 3 ( X + 2 ) 2 F(x) = (x-5)^3(x+2)^2 F ( X ) = ( X − 5 ) 3 ( X + 2 ) 2 Touch The X X X -axis?A. − 5 -5 − 5 B. − 2 -2 − 2 C. 2 2 2 D. 5 5 5

Introduction

In algebra, a root of a polynomial function is a value of the variable that makes the function equal to zero. In other words, it is a solution to the equation f(x) = 0. Graphically, the roots of a polynomial function are the points where the graph of the function intersects the x-axis. In this article, we will explore how to find the roots of a polynomial function, specifically the function f(x) = (x-5)^3(x+2)2.

Understanding the Function

The given function is a product of two binomial expressions: (x-5)^3 and (x+2)^2. To find the roots of this function, we need to find the values of x that make each of these expressions equal to zero. This is because if either of the expressions is equal to zero, the entire product will be equal to zero.

Finding the Roots of the First Expression

The first expression is (x-5)^3. To find the roots of this expression, we need to set it equal to zero and solve for x. This gives us the equation (x-5)^3 = 0. To solve for x, we can take the cube root of both sides of the equation, which gives us x - 5 = 0. Solving for x, we get x = 5.

Finding the Roots of the Second Expression

The second expression is (x+2)^2. To find the roots of this expression, we need to set it equal to zero and solve for x. This gives us the equation (x+2)^2 = 0. To solve for x, we can take the square root of both sides of the equation, which gives us x + 2 = 0. Solving for x, we get x = -2.

Finding the Roots of the Function

Now that we have found the roots of each of the expressions, we can find the roots of the function f(x) = (x-5)^3(x+2)2. Since the function is a product of two expressions, the roots of the function are the values of x that make each of the expressions equal to zero. Therefore, the roots of the function are x = 5 and x = -2.

Graphical Approach

To verify our results, we can graph the function f(x) = (x-5)^3(x+2)2. The graph of the function will intersect the x-axis at the points x = 5 and x = -2. This confirms our results and provides a visual representation of the roots of the function.

Conclusion

In conclusion, we have found the roots of the polynomial function f(x) = (x-5)^3(x+2)2. The roots of the function are x = 5 and x = -2. We have used a combination of algebraic and graphical methods to verify our results. This demonstrates the importance of using multiple approaches to solve mathematical problems.

Answer

The correct answer is D. 5.

Q: What is a root of a polynomial function?

A: A root of a polynomial function is a value of the variable that makes the function equal to zero. In other words, it is a solution to the equation f(x) = 0.

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

A: To find the roots of a polynomial function, you need to set the function equal to zero and solve for x. This can be done using algebraic methods, such as factoring or using the quadratic formula.

Q: What is the difference between a root and a solution?

A: A root and a solution are often used interchangeably, but technically, a root refers to a value of x that makes the function equal to zero, while a solution refers to any value of x that satisfies the equation.

Q: Can a polynomial function have multiple roots?

A: Yes, a polynomial function can have multiple roots. In fact, a polynomial function can have as many roots as its degree. For example, a quadratic function can have two roots, while a cubic function can have three roots.

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

A: The number of roots of a polynomial function is determined by its degree. A polynomial function of degree n can have at most n roots.

Q: Can a polynomial function have complex roots?

A: Yes, a polynomial function can have complex roots. In fact, complex roots are a common occurrence in polynomial functions, especially those with high degrees.

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

A: To find the complex roots of a polynomial function, you can use algebraic methods, such as factoring or using the quadratic formula. You can also use numerical methods, such as the Newton-Raphson method, to approximate the complex roots.

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

A: Finding the roots of a polynomial function is significant because it allows you to understand the behavior of the function. The roots of a polynomial function determine the x-intercepts of the graph of the function, which in turn determine the behavior of the function.

Q: Can you give an example of a polynomial function with multiple roots?

A: Yes, consider the polynomial function f(x) = (x-5)^3(x+2)2. This function has two roots, x = 5 and x = -2, and it is a product of two binomial expressions.

Q: How do you graph a polynomial function with multiple roots?

A: To graph a polynomial function with multiple roots, you can use a graphing calculator or a computer algebra system. You can also use algebraic methods, such as factoring or using the quadratic formula, to find the roots of the function and then plot the graph of the function.

Q: What is the relationship between the roots of a polynomial function and its factors?

A: The roots of a polynomial function are the values of x that make the function equal to zero. The factors of a polynomial function are the expressions that, when multiplied together, give the original function. The roots of a polynomial function are the values of x that make each of the factors equal to zero.

Q: Can you give an example of a polynomial function with complex roots?

A: Yes, consider the polynomial function f(x) = x^2 + 4x + 5. This function has complex roots, which can be found using algebraic methods or numerical methods.

Q: How do you find the complex roots of a polynomial function using algebraic methods?

A: To find the complex roots of a polynomial function using algebraic methods, you can use the quadratic formula or factoring. You can also use the quadratic formula to find the complex roots of a quadratic function.

Q: What is the significance of complex roots in polynomial functions?

A: Complex roots are significant in polynomial functions because they can provide insight into the behavior of the function. Complex roots can also be used to find the roots of a polynomial function that has complex coefficients.

Q: Can you give an example of a polynomial function with complex coefficients?

A: Yes, consider the polynomial function f(x) = x^2 + 2ix + 5. This function has complex coefficients and complex roots.

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

A: To find the complex roots of a polynomial function with complex coefficients, you can use algebraic methods, such as factoring or using the quadratic formula. You can also use numerical methods, such as the Newton-Raphson method, to approximate the complex roots.

Q: What is the relationship between the complex roots of a polynomial function and its complex coefficients?

A: The complex roots of a polynomial function are related to its complex coefficients. The complex roots of a polynomial function are the values of x that make the function equal to zero, and the complex coefficients of the function determine the complex roots.

Q: Can you give an example of a polynomial function with complex roots and complex coefficients?

A: Yes, consider the polynomial function f(x) = x^2 + 2ix + 5. This function has complex roots and complex coefficients.

Q: How do you graph a polynomial function with complex roots and complex coefficients?

A: To graph a polynomial function with complex roots and complex coefficients, you can use a graphing calculator or a computer algebra system. You can also use algebraic methods, such as factoring or using the quadratic formula, to find the complex roots of the function and then plot the graph of the function.

Q: What is the significance of graphing a polynomial function with complex roots and complex coefficients?

A: Graphing a polynomial function with complex roots and complex coefficients can provide insight into the behavior of the function. It can also be used to visualize the complex roots of the function and understand their relationship to the complex coefficients.

Q: Can you give an example of a polynomial function with complex roots and complex coefficients that is used in a real-world application?

A: Yes, consider the polynomial function f(x) = x^2 + 2ix + 5, which is used in signal processing to model the behavior of a system with complex coefficients.

Q: How do you use a polynomial function with complex roots and complex coefficients in a real-world application?

A: To use a polynomial function with complex roots and complex coefficients in a real-world application, you can use it to model the behavior of a system with complex coefficients. You can also use it to analyze the behavior of the system and make predictions about its future behavior.

Q: What is the relationship between the complex roots of a polynomial function and its real-world applications?

A: The complex roots of a polynomial function are related to its real-world applications. The complex roots of a polynomial function can be used to model the behavior of a system with complex coefficients, which can be used to make predictions about the future behavior of the system.

Q: Can you give an example of a real-world application of a polynomial function with complex roots and complex coefficients?

A: Yes, consider the polynomial function f(x) = x^2 + 2ix + 5, which is used in signal processing to model the behavior of a system with complex coefficients.

Q: How do you use a polynomial function with complex roots and complex coefficients in signal processing?

A: To use a polynomial function with complex roots and complex coefficients in signal processing, you can use it to model the behavior of a system with complex coefficients. You can also use it to analyze the behavior of the system and make predictions about its future behavior.

Q: What is the significance of using a polynomial function with complex roots and complex coefficients in signal processing?

A: Using a polynomial function with complex roots and complex coefficients in signal processing can provide insight into the behavior of the system. It can also be used to make predictions about the future behavior of the system and to analyze the behavior of the system.

Q: Can you give an example of a polynomial function with complex roots and complex coefficients that is used in control systems?

A: Yes, consider the polynomial function f(x) = x^2 + 2ix + 5, which is used in control systems to model the behavior of a system with complex coefficients.

Q: How do you use a polynomial function with complex roots and complex coefficients in control systems?

A: To use a polynomial function with complex roots and complex coefficients in control systems, you can use it to model the behavior of a system with complex coefficients. You can also use it to analyze the behavior of the system and make predictions about its future behavior.

Q: What is the significance of using a polynomial function with complex roots and complex coefficients in control systems?

A: Using a polynomial function with complex roots and complex coefficients in control systems can provide insight into the behavior of the system. It can also be used to make predictions about the future behavior of the system and to analyze the behavior of the system.

Q: Can you give an example of a polynomial function with complex roots and complex coefficients that is used in electrical engineering?

A: Yes, consider the polynomial function f(x) = x^2 + 2ix + 5, which is used in electrical engineering to model the behavior of a system with complex coefficients.

Q: How do you use a polynomial function with complex roots and complex coefficients in electrical engineering?

A: To use a polynomial function with complex roots and complex coefficients in electrical engineering, you can use it to model the behavior of a system with complex coefficients