Select The Correct Answer.Which Function Represents A Function With Zeros At − 3 , − 1 , 0 , -3, -1, 0, − 3 , − 1 , 0 , And 6 6 6 ?A. Y = ( X − 6 ) ( X + 1 ) ( X + 3 Y = (x-6)(x+1)(x+3 Y = ( X − 6 ) ( X + 1 ) ( X + 3 ] B. Y = X ( X − 3 ) ( X − 1 ) ( X + 6 Y = X(x-3)(x-1)(x+6 Y = X ( X − 3 ) ( X − 1 ) ( X + 6 ] C. Y = X ( X − 6 ) ( X + 1 ) ( X + 3 Y = X(x-6)(x+1)(x+3 Y = X ( X − 6 ) ( X + 1 ) ( X + 3 ] D. $y =

Understanding the Problem

In this problem, we are tasked with identifying the function that represents a function with zeros at $-3, -1, 0,$ and $6$. This means that the function should have roots or x-intercepts at these specific points. To solve this problem, we need to analyze each option and determine which one satisfies the given conditions.

Analyzing Option A

Option A is given by the function $y = (x-6)(x+1)(x+3)$. Let's examine this function to see if it meets the criteria.

• The function has a root at $x = 6$ because when $x = 6$, the function becomes $y = (6-6)(6+1)(6+3) = 0$.
• The function has a root at $x = -1$ because when $x = -1$, the function becomes $y = (-1-6)(-1+1)(-1+3) = 0$.
• The function has a root at $x = -3$ because when $x = -3$, the function becomes $y = (-3-6)(-3+1)(-3+3) = 0$.
• However, the function does not have a root at $x = 0$ because when $x = 0$, the function becomes $y = (0-6)(0+1)(0+3) = 18$.

Analyzing Option B

Option B is given by the function $y = x(x-3)(x-1)(x+6)$. Let's examine this function to see if it meets the criteria.

• The function has a root at $x = 0$ because when $x = 0$, the function becomes $y = 0(0-3)(0-1)(0+6) = 0$.
• The function has a root at $x = 3$ because when $x = 3$, the function becomes $y = 3(3-3)(3-1)(3+6) = 0$.
• The function has a root at $x = 1$ because when $x = 1$, the function becomes $y = 1(1-3)(1-1)(1+6) = 0$.
• However, the function does not have a root at $x = -6$ because when $x = -6$, the function becomes $y = -6(-6-3)(-6-1)(-6+6) = 0$.

Analyzing Option C

Option C is given by the function $y = x(x-6)(x+1)(x+3)$. Let's examine this function to see if it meets the criteria.

• The function has a root at $x = 0$ because when $x = 0$, the function becomes $y = 0(0-6)(0+1)(0+3) = 0$.
• The function has a root at $x = 6$ because when $x = 6$, the function becomes $y = 6(6-6)(6+1)(6+3) = 0$.
• The function has a root at $x = -1$ because when $x = -1$, the function becomes $y = -1(-1-6)(-1+1)(-1+3) = 0$.
• The function has a root at $x = -3$ because when $x = -3$, the function becomes $y = -3(-3-6)(-3+1)(-3+3) = 0$.

Conclusion

Based on the analysis of each option, we can conclude that the function that represents a function with zeros at $-3, -1, 0,$ and $6$ is Option C.

The Correct Answer

The correct answer is:

• C. $y = x(x-6)(x+1)(x+3)$

Frequently Asked Questions

In the previous article, we discussed how to select the correct answer for a function with zeros at $-3, -1, 0,$ and $6$. However, we understand that there may be more questions and concerns regarding this topic. In this article, we will address some of the most frequently asked questions and provide additional insights to help you better understand functions with zeros.

Q: What is a function with zeros?

A function with zeros is a function that has roots or x-intercepts at specific points. In other words, when the function is equal to zero, the x-value of the point is called a root or zero of the function.

Q: How do I determine if a function has a zero at a specific point?

To determine if a function has a zero at a specific point, you can substitute the x-value of the point into the function and check if the result is equal to zero. If the result is zero, then the function has a zero at that point.

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

A root and a zero are often used interchangeably, but technically, a root is the x-value of the point where the function is equal to zero, while a zero is the function itself at that point.

Q: Can a function have multiple zeros?

Yes, a function can have multiple zeros. In fact, the function we discussed earlier, $y = x(x-6)(x+1)(x+3)$, has four zeros at $x = -3, -1, 0,$ and $6$.

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

There are several methods to find the zeros of a function, including:

• Factoring: If the function can be factored into the product of linear factors, you can set each factor equal to zero and solve for the x-value.
• Graphing: You can graph the function and look for the x-intercepts, which represent the zeros of the function.
• Numerical methods: You can use numerical methods, such as the Newton-Raphson method, to approximate the zeros of the function.

Q: What is the significance of zeros in a function?

Zeros in a function are significant because they represent the points where the function changes sign. In other words, if a function has a zero at a point, it means that the function changes from positive to negative or vice versa at that point.

Q: Can a function have complex zeros?

Yes, a function can have complex zeros. Complex zeros are zeros that have a non-zero imaginary part. In other words, if a function has a zero at a complex number, it means that the function changes sign at that complex number.

Conclusion

In this article, we addressed some of the most frequently asked questions regarding functions with zeros. We hope that this article has provided you with a better understanding of functions with zeros and how to determine if a function has a zero at a specific point. If you have any further questions or concerns, please don't hesitate to ask.

Additional Resources

For more information on functions with zeros, we recommend the following resources:

• Khan Academy: Functions with Zeros
• Mathway: Functions with Zeros
• Wolfram Alpha: Functions with Zeros

We hope that this article has been helpful in your understanding of functions with zeros. If you have any further questions or concerns, please don't hesitate to ask.