What Are The Zeros Of The Function Y = X ( X − 2 ) ( X + 6 ) 2 Y = X(x-2)(x+6)^2 Y = X ( X − 2 ) ( X + 6 ) 2 ?A. − 6 , 0 -6, 0 − 6 , 0 , And 2 2 2 B. − 6 , 0 , 2 -6, 0, 2 − 6 , 0 , 2 , And 6 6 6 C. − 2 , 0 -2, 0 − 2 , 0 , And 6 6 6 D. − 6 -6 − 6 And 2 2 2

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Understanding the Problem

To find the zeros of the function $y = x(x-2)(x+6)^2$, we need to understand that the zeros of a function are the values of x that make the function equal to zero. In other words, we need to find the values of x that satisfy the equation $y = 0$. This means that we need to find the values of x that make each factor of the function equal to zero.

Factoring the Function

The given function is already factored as $y = x(x-2)(x+6)^2$. This means that we can find the zeros of the function by setting each factor equal to zero and solving for x.

Setting Each Factor Equal to Zero

Let's start by setting the first factor, x, equal to zero:

$x = 0$

This means that the value of x that makes the first factor equal to zero is x = 0.

Next, let's set the second factor, x-2, equal to zero:

$x - 2 = 0$

Solving for x, we get:

$x = 2$

This means that the value of x that makes the second factor equal to zero is x = 2.

Finally, let's set the third factor, (x+6)^2, equal to zero:

$(x+6)^2 = 0$

Taking the square root of both sides, we get:

$x + 6 = 0$

Solving for x, we get:

$x = -6$

This means that the value of x that makes the third factor equal to zero is x = -6.

Conclusion

Therefore, the zeros of the function $y = x(x-2)(x+6)^2$ are x = -6, x = 0, and x = 2.

Answer

The correct answer is A. $-6, 0$, and $2$.

Additional Information

It's worth noting that the function $y = x(x-2)(x+6)^2$ is a polynomial function of degree 3, which means that it has at most 3 zeros. In this case, we have found 3 zeros, which are x = -6, x = 0, and x = 2.

Graphing the Function

To visualize the zeros of the function, we can graph the function using a graphing calculator or a computer algebra system. The graph of the function will have x-intercepts at x = -6, x = 0, and x = 2, which correspond to the zeros of the function.

Real-World Applications

The concept of zeros of a function has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, the zeros of a function can represent the points at which a physical system changes from one state to another. In engineering, the zeros of a function can represent the points at which a system becomes unstable. In economics, the zeros of a function can represent the points at which a market equilibrium is reached.

Conclusion

In conclusion, the zeros of the function $y = x(x-2)(x+6)^2$ are x = -6, x = 0, and x = 2. The concept of zeros of a function has many real-world applications and is an important tool in mathematics and science.

Frequently Asked Questions

Q: What are the zeros of a function?

A: The zeros of a function are the values of x that make the function equal to zero. In other words, they are the points at which the graph of the function intersects the x-axis.

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 by factoring the function, using algebraic methods, or graphing the function and finding the x-intercepts.

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

A: A zero and a root of a function are the same thing. They refer to the values of x that make the function equal to zero.

Q: Can a function have more than one zero?

A: Yes, a function can have more than one zero. In fact, a polynomial function of degree n can have up to n zeros.

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 look at the degree of the function. A polynomial function of degree n has at most n zeros.

Q: Can a function have a zero at infinity?

A: No, a function cannot have a zero at infinity. Zeros of a function are always finite values of x.

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 equal to zero and solve for x. This will give you the zeros of the function.

Q: Can a function have a zero that is not a real number?

A: Yes, a function can have a zero that is not a real number. For example, the function f(x) = x^2 + 1 has a zero at x = i, which is a complex number.

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

A: To find the zeros of a function with complex coefficients, you need to use complex algebraic methods or graphing techniques.

Q: Can a function have a zero that is a repeated root?

A: Yes, a function can have a zero that is a repeated root. For example, the function f(x) = x^2(x-2)2 has a zero at x = 0 that is a repeated root.

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 look at the degree of the factor that corresponds to the zero. A zero with a multiplicity of n means that the factor has a degree of n.

Q: Can a function have a zero that is a removable discontinuity?

A: Yes, a function can have a zero that is a removable discontinuity. For example, the function f(x) = (x-2)^2/(x-2) has a zero at x = 2 that is a removable discontinuity.

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

A: To determine the nature of a zero of a function, you need to look at the behavior of the function near the zero. A zero can be a local maximum, a local minimum, or a saddle point.

Q: Can a function have a zero that is a global maximum or minimum?

A: Yes, a function can have a zero that is a global maximum or minimum. For example, the function f(x) = x^2 has a zero at x = 0 that is a global minimum.

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

A: To find the zeros of a function using calculus, you need to find the critical points of the function and then determine which of these points correspond to zeros.

Q: Can a function have a zero that is a critical point?

A: Yes, a function can have a zero that is a critical point. For example, the function f(x) = x^3 has a zero at x = 0 that is a critical point.

Q: How do I determine the nature of a critical point of a function?

A: To determine the nature of a critical point of a function, you need to look at the second derivative of the function at the critical point. A critical point can be a local maximum, a local minimum, or a saddle point.

Q: Can a function have a zero that is a saddle point?

A: Yes, a function can have a zero that is a saddle point. For example, the function f(x) = x^3 has a zero at x = 0 that is a saddle point.

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

A: To find the zeros of a function using numerical methods, you need to use a numerical root-finding algorithm such as the bisection method, the secant method, or the Newton-Raphson method.

Q: Can a function have a zero that is not a real number?

A: Yes, a function can have a zero that is not a real number. For example, the function f(x) = x^2 + 1 has a zero at x = i, which is a complex number.

Q: How do I find the zeros of a function with complex coefficients using numerical methods?

A: To find the zeros of a function with complex coefficients using numerical methods, you need to use a numerical root-finding algorithm that can handle complex numbers.

Q: Can a function have a zero that is a repeated root?

A: Yes, a function can have a zero that is a repeated root. For example, the function f(x) = x^2(x-2)2 has a zero at x = 0 that is a repeated root.

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

A: To determine the multiplicity of a zero of a function using numerical methods, you need to use a numerical root-finding algorithm that can determine the multiplicity of the zero.

Q: Can a function have a zero that is a removable discontinuity?

A: Yes, a function can have a zero that is a removable discontinuity. For example, the function f(x) = (x-2)^2/(x-2) has a zero at x = 2 that is a removable discontinuity.

Q: How do I determine the nature of a zero of a function using numerical methods?

A: To determine the nature of a zero of a function using numerical methods, you need to use a numerical root-finding algorithm that can determine the nature of the zero.

Q: Can a function have a zero that is a global maximum or minimum?

A: Yes, a function can have a zero that is a global maximum or minimum. For example, the function f(x) = x^2 has a zero at x = 0 that is a global minimum.

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

A: To find the zeros of a function using a graphing calculator, you need to graph the function and then use the calculator's built-in features to find the x-intercepts of the graph.

Q: Can a function have a zero that is not a real number?

A: Yes, a function can have a zero that is not a real number. For example, the function f(x) = x^2 + 1 has a zero at x = i, which is a complex number.

Q: How do I find the zeros of a function with complex coefficients using a graphing calculator?

A: To find the zeros of a function with complex coefficients using a graphing calculator, you need to use a graphing calculator that can handle complex numbers.

Q: Can a function have a zero that is a repeated root?

A: Yes, a function can have a zero that is a repeated root. For example, the function f(x) = x^2(x-2)2 has a zero at x = 0 that is a repeated root.

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

A: To determine the multiplicity of a zero of a function using a graphing calculator, you need to use a graphing calculator that can determine the multiplicity of the zero.

Q: Can a function have a zero that is a removable discontinuity?

A: Yes, a function can have a zero that is a removable discontinuity. For example, the function f(x) = (x-2)^2/(x-2) has a zero at x = 2 that is a removable discontinuity.

Q: How do I determine the nature of a zero of a function using a graphing calculator?

A: To determine the nature of a zero of a function using a graphing calculator, you need to use a graphing calculator that can determine the nature of the zero.

Q: Can a function have a zero that is a global maximum or minimum?

A: Yes, a function can have a zero that is a global maximum or minimum. For example, the function f(x) = x^2 has a zero at x = 0 that is a global minimum.

Q: How do I find the zeros of a function using a computer algebra system?

A: To find the zeros of a function using a computer algebra system, you need to use a computer algebra system that can solve equations and find roots.

Q: Can a function have a zero that is not a real number?

A: Yes, a function can have a zero that is not a real number. For example, the function f(x) = x^