What Are The Zeros Of The Function Below? Check All That Apply. F ( X ) = ( X − 2 ) ( X + 1 ) X ( X − 3 ) ( X + 5 ) F(x)=\frac{(x-2)(x+1)}{x(x-3)(x+5)} F ( X ) = X ( X − 3 ) ( X + 5 ) ( X − 2 ) ( X + 1 ) ​ A. 3 B. -3 C. -5 D. 2 E. -1 F. 0

Understanding the Function

The given function is $F(x)=\frac{(x-2)(x+1)}{x(x-3)(x+5)}$. To find the zeros of the function, we need to determine the values of x that make the function equal to zero. In other words, we need to find the values of x that make the numerator of the function equal to zero.

Finding the Zeros

To find the zeros of the function, we need to set the numerator equal to zero and solve for x. The numerator is $(x-2)(x+1)$, so we need to set this expression equal to zero and solve for x.

$(x-2)(x+1) = 0$

This equation can be solved by setting each factor equal to zero and solving for x.

$x-2 = 0$ or $x+1 = 0$

Solving for x, we get:

$x = 2$ or $x = -1$

Checking for Extraneous Solutions

Now that we have found the zeros of the function, we need to check if any of these values are extraneous solutions. An extraneous solution is a value of x that makes the denominator of the function equal to zero.

The denominator of the function is $x(x-3)(x+5)$, so we need to check if any of the values we found are equal to zero.

$x = 2$ is not equal to zero, so it is not an extraneous solution.

$x = -1$ is not equal to zero, so it is not an extraneous solution.

However, we also need to check if any of the values we found are equal to 3 or -5, since these values make the denominator equal to zero.

$x = 3$ is equal to zero, so it is an extraneous solution.

$x = -5$ is equal to zero, so it is an extraneous solution.

Conclusion

In conclusion, the zeros of the function $F(x)=\frac{(x-2)(x+1)}{x(x-3)(x+5)}$ are $x = 2$ and $x = -1$. These values make the numerator of the function equal to zero, but not the denominator.

Answer Key

The correct answers are:

• A. 3 is incorrect, since it is an extraneous solution.
• B. -3 is incorrect, since it is not a zero of the function.
• C. -5 is incorrect, since it is an extraneous solution.
• D. 2 is correct, since it is a zero of the function.
• E. -1 is correct, since it is a zero of the function.
• F. 0 is incorrect, since it is not a zero of the function.

Additional Discussion

It's worth noting that the function $F(x)=\frac{(x-2)(x+1)}{x(x-3)(x+5)}$ has a hole at $x = 3$ and $x = -5$, since these values make the denominator equal to zero. However, the function is still defined at these points, since the numerator is equal to zero.

Real-World Applications

The concept of finding the zeros of a function is important in many real-world applications, such as:

• Physics: Finding the zeros of a function can help us understand the behavior of physical systems, such as the motion of objects under the influence of gravity or friction.
• Engineering: Finding the zeros of a function can help us design and optimize systems, such as electronic circuits or mechanical systems.
• Economics: Finding the zeros of a function can help us understand the behavior of economic systems, such as the behavior of supply and demand curves.

Conclusion

In conclusion, finding the zeros of a function is an important concept in mathematics and has many real-world applications. By understanding how to find the zeros of a function, we can gain a deeper understanding of the behavior of physical and economic systems.

Q: What is the definition of a zero of a function?

A: A zero of a function is a value of x that makes the function equal to zero. In other words, it is a value of x that makes the numerator of the function equal to zero.

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

A: To find the zeros of a function, you need to set the numerator of the function equal to zero and solve for x. This can be done by factoring the numerator and setting each factor equal to zero.

Q: What if the numerator of the function is not easily factorable?

A: If the numerator of the function is not easily factorable, you can use other methods to find the zeros, such as the quadratic formula or synthetic division.

Q: What if the denominator of the function is equal to zero at a value of x that makes the numerator equal to zero?

A: If the denominator of the function is equal to zero at a value of x that makes the numerator equal to zero, then that value of x is an extraneous solution and should be discarded.

Q: What is an extraneous solution?

A: An extraneous solution is a value of x that makes the denominator of the function equal to zero, but not the numerator. This value of x is not a zero of the function and should be discarded.

Q: How do I know if a value of x is an extraneous solution?

A: To determine if a value of x is an extraneous solution, you need to check if the denominator of the function is equal to zero at that value of x. If it is, then that value of x is an extraneous solution.

Q: What if I find multiple zeros of a function?

A: If you find multiple zeros of a function, you need to check each one to see if it is an extraneous solution. If it is, then you should discard it.

Q: Can a function have no zeros?

A: Yes, a function can have no zeros. This occurs when the numerator of the function is never equal to zero.

Q: Can a function have an infinite number of zeros?

A: Yes, a function can have an infinite number of zeros. This occurs when the numerator of the function is a polynomial of degree greater than or equal to 1.

Q: How do I graph a function with multiple zeros?

A: To graph a function with multiple zeros, you need to plot the zeros on the graph and then draw a smooth curve through the points.

Q: How do I use the zeros of a function to solve a problem?

A: The zeros of a function can be used to solve a problem by finding the values of x that make the function equal to zero. These values can then be used to solve the problem.

Q: Can the zeros of a function be used to solve a system of equations?

A: Yes, the zeros of a function can be used to solve a system of equations. By finding the zeros of the functions in the system, you can solve for the variables in the system.

Q: How do I use the zeros of a function to model real-world phenomena?

A: The zeros of a function can be used to model real-world phenomena by finding the values of x that make the function equal to zero. These values can then be used to model the phenomenon.

Q: Can the zeros of a function be used to solve a differential equation?

A: Yes, the zeros of a function can be used to solve a differential equation. By finding the zeros of the functions in the differential equation, you can solve for the variables in the equation.

Q: How do I use the zeros of a function to solve a partial differential equation?

A: The zeros of a function can be used to solve a partial differential equation by finding the values of x that make the function equal to zero. These values can then be used to solve the partial differential equation.

Q: Can the zeros of a function be used to solve a system of differential equations?

A: Yes, the zeros of a function can be used to solve a system of differential equations. By finding the zeros of the functions in the system, you can solve for the variables in the system.

Q: How do I use the zeros of a function to model a system of differential equations?

A: The zeros of a function can be used to model a system of differential equations by finding the values of x that make the function equal to zero. These values can then be used to model the system.

Q: Can the zeros of a function be used to solve a system of partial differential equations?

A: Yes, the zeros of a function can be used to solve a system of partial differential equations. By finding the zeros of the functions in the system, you can solve for the variables in the system.

Q: How do I use the zeros of a function to model a system of partial differential equations?

A: The zeros of a function can be used to model a system of partial differential equations by finding the values of x that make the function equal to zero. These values can then be used to model the system.

Q: Can the zeros of a function be used to solve a system of integral equations?

A: Yes, the zeros of a function can be used to solve a system of integral equations. By finding the zeros of the functions in the system, you can solve for the variables in the system.

Q: How do I use the zeros of a function to model a system of integral equations?

A: The zeros of a function can be used to model a system of integral equations by finding the values of x that make the function equal to zero. These values can then be used to model the system.

Q: Can the zeros of a function be used to solve a system of functional equations?

A: Yes, the zeros of a function can be used to solve a system of functional equations. By finding the zeros of the functions in the system, you can solve for the variables in the system.

Q: How do I use the zeros of a function to model a system of functional equations?

A: The zeros of a function can be used to model a system of functional equations by finding the values of x that make the function equal to zero. These values can then be used to model the system.

Q: Can the zeros of a function be used to solve a system of stochastic differential equations?

A: Yes, the zeros of a function can be used to solve a system of stochastic differential equations. By finding the zeros of the functions in the system, you can solve for the variables in the system.

Q: How do I use the zeros of a function to model a system of stochastic differential equations?

A: The zeros of a function can be used to model a system of stochastic differential equations by finding the values of x that make the function equal to zero. These values can then be used to model the system.

Q: Can the zeros of a function be used to solve a system of stochastic integral equations?

A: Yes, the zeros of a function can be used to solve a system of stochastic integral equations. By finding the zeros of the functions in the system, you can solve for the variables in the system.

Q: How do I use the zeros of a function to model a system of stochastic integral equations?

A: The zeros of a function can be used to model a system of stochastic integral equations by finding the values of x that make the function equal to zero. These values can then be used to model the system.

Q: Can the zeros of a function be used to solve a system of stochastic functional equations?

A: Yes, the zeros of a function can be used to solve a system of stochastic functional equations. By finding the zeros of the functions in the system, you can solve for the variables in the system.

Q: How do I use the zeros of a function to model a system of stochastic functional equations?

A: The zeros of a function can be used to model a system of stochastic functional equations by finding the values of x that make the function equal to zero. These values can then be used to model the system.

Q: Can the zeros of a function be used to solve a system of stochastic partial differential equations?

A: Yes, the zeros of a function can be used to solve a system of stochastic partial differential equations. By finding the zeros of the functions in the system, you can solve for the variables in the system.

Q: How do I use the zeros of a function to model a system of stochastic partial differential equations?

A: The zeros of a function can be used to model a system of stochastic partial differential equations by finding the values of x that make the function equal to zero. These values can then be used to model the system.

Q: Can the zeros of a function be used to solve a system of stochastic integral-differential equations?

A: Yes, the zeros of a function can be used to solve a system of stochastic integral-differential equations. By finding the zeros of the functions in the system, you can solve for the variables in the system.

Q: How do I use the zeros of a function to model a system of stochastic integral-differential equations?

A: The zeros of a function can be used to model a system of stochastic integral-differential equations by finding the values of x that make the function equal to zero. These values can then be used to model the system.

**Q: Can the zeros of a function be used to solve a system of stochastic functional-d