Find The Zeros Of The Function And State Their Multiplicities.$ C(x) = 6x^3 + 9x^2 - 60x }$If There Is More Than One Answer, Separate Them With Commas. Select None If Applicable.Zero(s) Of { C $}$ { \square, \square$ $
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Introduction
In algebra, finding the zeros of a function is a crucial step in understanding its behavior and properties. A zero of a function is a value of the variable that makes the function equal to zero. In this article, we will focus on finding the zeros of a cubic function, which is a polynomial function of degree three. We will use the given function c(x) = 6x^3 + 9x^2 - 60x as an example and follow a step-by-step approach to find its zeros and state their multiplicities.
Understanding the Function
Before we start finding the zeros, let's take a closer look at the given function c(x) = 6x^3 + 9x^2 - 60x. This is a cubic function, which means it has a degree of three. The general form of a cubic function is f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
Factoring the Function
To find the zeros of the function, we can start by factoring it. Factoring a cubic function can be challenging, but we can try to factor out any common factors. In this case, we can factor out a 3x from the function:
c(x) = 6x^3 + 9x^2 - 60x = 3x(2x^2 + 3x - 20)
Finding the Zeros
Now that we have factored the function, we can find its zeros by setting each factor equal to zero and solving for x. Let's start with the first factor, 3x:
3x = 0 x = 0
This is one of the zeros of the function, and its multiplicity is 1.
Solving the Quadratic Factor
Next, we need to solve the quadratic factor 2x^2 + 3x - 20. We can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 2, b = 3, and c = -20. Plugging these values into the formula, we get:
x = (-(3) ± √((3)^2 - 4(2)(-20))) / 2(2) x = (-3 ± √(9 + 160)) / 4 x = (-3 ± √169) / 4 x = (-3 ± 13) / 4
This gives us two possible values for x:
x = (-3 + 13) / 4 = 10/4 = 2.5 x = (-3 - 13) / 4 = -16/4 = -4
Finding the Multiplicities
Now that we have found the zeros of the function, we need to determine their multiplicities. The multiplicity of a zero is the number of times the factor corresponding to that zero appears in the factored form of the function.
In this case, we have three zeros: x = 0, x = 2.5, and x = -4. The factor corresponding to x = 0 is 3x, which appears only once in the factored form of the function. Therefore, the multiplicity of x = 0 is 1.
The factors corresponding to x = 2.5 and x = -4 are 2x - 13 and 2x + 13, respectively. Both of these factors appear only once in the factored form of the function. Therefore, the multiplicities of x = 2.5 and x = -4 are both 1.
Conclusion
In conclusion, we have found the zeros of the given cubic function c(x) = 6x^3 + 9x^2 - 60x and stated their multiplicities. The zeros are x = 0, x = 2.5, and x = -4, and their multiplicities are 1, 1, and 1, respectively.
Final Answer
The final answer is:
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Introduction
In our previous article, we explored the concept of finding the zeros of a cubic function. We used the function c(x) = 6x^3 + 9x^2 - 60x as an example and followed a step-by-step approach to find its zeros and state their multiplicities. In this article, we will answer some frequently asked questions about cubic function zeros.
Q&A
Q: What is a cubic function?
A: A cubic function is a polynomial function of degree three. It has the general form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
Q: How do I find the zeros of a cubic function?
A: To find the zeros of a cubic function, you can start by factoring the function. If the function can be factored, you can set each factor equal to zero and solve for x. If the function cannot be factored, you can use the quadratic formula to solve for x.
Q: What is the multiplicity of a zero?
A: The multiplicity of a zero is the number of times the factor corresponding to that zero appears in the factored form of the function.
Q: How do I determine the multiplicity of a zero?
A: To determine the multiplicity of a zero, you need to look at the factored form of the function and count the number of times the factor corresponding to that zero appears.
Q: Can a cubic function have more than one zero?
A: Yes, a cubic function can have more than one zero. In fact, a cubic function can have up to three zeros.
Q: How do I know if a zero is repeated?
A: If a zero is repeated, it means that the factor corresponding to that zero appears more than once in the factored form of the function.
Q: Can a zero be a complex number?
A: Yes, a zero can be a complex number. In fact, complex numbers can be zeros of a cubic function.
Q: How do I find the zeros of a cubic function with complex coefficients?
A: To find the zeros of a cubic function with complex coefficients, you can use the quadratic formula and the fact that complex numbers can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit.
Q: What is the significance of finding the zeros of a cubic function?
A: Finding the zeros of a cubic function is important because it helps us understand the behavior of the function and its graph. The zeros of a function are the points where the function intersects the x-axis, and they can provide valuable information about the function's properties.
Conclusion
In conclusion, finding the zeros of a cubic function is an important step in understanding the behavior of the function and its graph. By following the steps outlined in this article, you can find the zeros of a cubic function and determine their multiplicities.
Final Answer
The final answer is: The zeros of a cubic function are the points where the function intersects the x-axis, and they can provide valuable information about the function's properties.