Understanding the Concept of Zeros

Oct 13, 2025 by windailys.com 80 views

In mathematics, a function with zeros at specific points is often represented by a polynomial equation. The zeros of a function are the values of x that make the function equal to zero. In this article, we will explore how to select the correct function with zeros at $-3, -1, 0$ , and $6$ .

Understanding the Concept of Zeros

The zeros of a function are the values of x that make the function equal to zero. In other words, if a function f(x) has a zero at x = a, then f(a) = 0. The zeros of a function can be found by setting the function equal to zero and solving for x.

Representing Functions with Zeros

A function with zeros at specific points can be represented by a polynomial equation. The polynomial equation can be written in the form:

y = (x - a)(x - b)(x - c)...(x - n)

where a, b, c, ..., n are the zeros of the function.

Selecting the Correct Function

Now, let's select the correct function with zeros at $-3, -1, 0$ , and $6$ . We need to find the polynomial equation that represents this function.

Option A: y = (x - 6)(x + 1)(x + 3)

This option represents a function with zeros at x = 6, x = -1, and x = -3. However, this function does not have a zero at x = 0.

Option B: y = x(x - 3)(x - 1)(x + 6)

This option represents a function with zeros at x = 0, x = 3, x = 1, and x = -6. However, this function does not have zeros at x = -3 and x = -1.

Option C: y = x(x - 6)(x + 1)(x + 3)

This option represents a function with zeros at x = 0, x = 6, x = -1, and x = -3. This is the correct function with zeros at $-3, -1, 0$ , and $6$ .

Option D: y = (x - 3)(x - 1)(x + 6)

This option represents a function with zeros at x = 3, x = 1, and x = -6. However, this function does not have zeros at x = -3, x = -1, and x = 0.

Conclusion

In conclusion, the correct function with zeros at $-3, -1, 0$ , and $6$ is represented by the polynomial equation:

y = x(x - 6)(x + 1)(x + 3)

This function has zeros at x = 0, x = 6, x = -1, and x = -3, which matches the given zeros.

Key Takeaways

A function with zeros at specific points can be represented by a polynomial equation.
The polynomial equation can be written in the form: y = (x - a)(x - b)(x - c)...(x - n)
The zeros of a function are the values of x that make the function equal to zero.
The correct function with zeros at $-3, -1, 0$ , and $6$ is represented by the polynomial equation: y = x(x - 6)(x + 1)(x + 3)
Frequently Asked Questions (FAQs) about Functions with Zeros ====================================================================

In the previous article, we discussed how to select the correct function with zeros at $-3, -1, 0$ , and $6$ . In this article, we will answer some frequently asked questions (FAQs) about functions with zeros.

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

A: A zero of a function is a value of x that makes the function equal to zero. A root of a function is a value of x that makes the function equal to zero, but it can also be a value of x that makes the function equal to a non-zero constant.

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

A: To find the zeros of a 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 relationship between the zeros of a function and its graph?

A: The zeros of a function are the x-intercepts of its graph. In other words, if a function has a zero at x = a, then the graph of the function will cross the x-axis at x = a.

Q: Can a function have more than one zero?

A: Yes, a function can have more than one zero. For example, the function y = x(x - 1)(x - 2) has zeros at x = 0, x = 1, and x = 2.

Q: How do I determine the number of zeros of a function?

A: To determine the number of zeros of a function, you need to examine the degree of the function. If the degree of the function is even, then the function can have an even number of zeros. If the degree of the function is odd, then the function can have an odd number of zeros.

Q: Can a function have a zero at x = 0?

A: Yes, a function can have a zero at x = 0. For example, the function y = x(x - 1)(x - 2) has a zero at x = 0.

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

A: To find the zeros of a rational function, you need to set the numerator of 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: Can a function have a zero at x = infinity?

A: No, a function cannot have a zero at x = infinity. A zero of a function is a value of x that makes the function equal to zero, and x = infinity is not a value of x.

Q: How do I determine the multiplicity of a zero of a function?

A: To determine the multiplicity of a zero of a function, you need to examine the factorization of the function. If the function can be factored into the form y = (x - a)^n, then the zero at x = a has a multiplicity of n.

Conclusion

In conclusion, functions with zeros are an important concept in mathematics. By understanding how to find the zeros of a function and how to determine the number of zeros, you can gain a deeper understanding of the behavior of functions and their graphs.

Key Takeaways

A zero of a function is a value of x that makes the function equal to zero.
A root of a function is a value of x that makes the function equal to zero, but it can also be a value of x that makes the function equal to a non-zero constant.
The zeros of a function are the x-intercepts of its graph.
A function can have more than one zero.
The degree of a function determines the number of zeros it can have.
A function can have a zero at x = 0.
A rational function can have zeros at the values of x that make the numerator equal to zero.
A function cannot have a zero at x = infinity.
The multiplicity of a zero of a function can be determined by examining the factorization of the function.