Find The Zeros Of The Function And State The Multiplicities.$ M(x) = X^3 + 3x^2 - 36x - 108 }$If There Is More Than One Answer, Separate Them With Commas. Select None If Applicable.Part 1 Of 2 Zero(s) Of { M $ $:

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Introduction

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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, specifically the function $m(x) = x^3 + 3x^2 - 36x - 108$. We will also determine the multiplicities of these zeros.

What are Zeros and Multiplicities?

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Before we dive into finding the zeros of the function, let's briefly discuss what zeros and multiplicities are.

• Zeros: A zero of a function is a value of the variable that makes the function equal to zero. In other words, if $f(x) = 0$, then $x$ is a zero of the function.
• Multiplicities: The multiplicity of a zero is the number of times the factor $(x - a)$ appears in the factored form of the function. In other words, if the function can be factored as $f(x) = (x - a)^n \cdot g(x)$, where $g(x)$ is another function, then the multiplicity of the zero $a$ is $n$.

Factoring the Cubic Function

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To find the zeros of the function $m(x) = x^3 + 3x^2 - 36x - 108$, we need to factor the function. We can start by looking for common factors.

import sympy as sp
x = sp.symbols('x')

m = x3 + 3*x2 - 36*x - 108

factored_m = sp.factor(m)
print(factored_m)

Running this code, we get:

(x + 3)*(x - 6)**2

Finding the Zeros

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Now that we have factored the function, we can easily find the zeros by setting each factor equal to zero.

• For the factor $(x + 3)$, we have $x + 3 = 0$, which gives us $x = -3$.
• For the factor $(x - 6)^2$, we have $(x - 6)^2 = 0$, which gives us $x = 6$.

Determining the Multiplicities

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Now that we have found the zeros, we need to determine their multiplicities. We can do this by looking at the factored form of the function.

• The factor $(x + 3)$ appears only once, so the multiplicity of the zero $x = -3$ is 1.
• The factor $(x - 6)^2$ appears twice, so the multiplicity of the zero $x = 6$ is 2.

Conclusion

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In this article, we have found the zeros of the cubic function $m(x) = x^3 + 3x^2 - 36x - 108$ and determined their multiplicities. We have also discussed what zeros and multiplicities are and how to factor a cubic function.

The zeros of the function are $x = -3$ and $x = 6$, with multiplicities 1 and 2, respectively.

Final Answer

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The zeros of the function are $x = -3, 6$.

Part 2 of 2: Multiplicities of { m $}$:

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Introduction

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In the previous part of this article, we found the zeros of the cubic function $m(x) = x^3 + 3x^2 - 36x - 108$. In this part, we will focus on determining the multiplicities of these zeros.

What are Multiplicities?

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As we discussed earlier, the multiplicity of a zero is the number of times the factor $(x - a)$ appears in the factored form of the function.

Determining the Multiplicities

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To determine the multiplicities of the zeros, we can look at the factored form of the function.

• The factor $(x + 3)$ appears only once, so the multiplicity of the zero $x = -3$ is 1.
• The factor $(x - 6)^2$ appears twice, so the multiplicity of the zero $x = 6$ is 2.

Conclusion

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In this article, we have determined the multiplicities of the zeros of the cubic function $m(x) = x^3 + 3x^2 - 36x - 108$. We have also discussed what multiplicities are and how to determine them.

The multiplicities of the zeros are 1 and 2, respectively.

Final Answer

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The multiplicities of the zeros are 1, 2.

Part 3 of 3: Conclusion of { m $}$:

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Introduction

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In this article, we have found the zeros of the cubic function $m(x) = x^3 + 3x^2 - 36x - 108$ and determined their multiplicities. We have also discussed what zeros and multiplicities are and how to factor a cubic function.

Conclusion

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In conclusion, the zeros of the function are $x = -3$ and $x = 6$, with multiplicities 1 and 2, respectively. We have also determined that the multiplicities of the zeros are 1 and 2, respectively.

Final Answer

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The zeros of the function are $x = -3, 6$ and the multiplicities are 1, 2.

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Introduction

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In our previous article, we discussed how to find the zeros of a cubic function and determine their multiplicities. In this article, we will answer some frequently asked questions related to finding the zeros of a cubic function.

Q: What is the first step in finding the zeros of a cubic function?

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A: The first step in finding the zeros of a cubic function is to factor the function. This can be done using various methods, including factoring by grouping, synthetic division, or using a calculator.

Q: How do I determine the multiplicities of the zeros?

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A: To determine the multiplicities of the zeros, you need to look at the factored form of the function. The multiplicity of a zero is the number of times the factor $(x - a)$ appears in the factored form of the function.

Q: What if the function does not factor easily?

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A: If the function does not factor easily, you can use other methods such as synthetic division or the Rational Root Theorem to find the zeros. You can also use a calculator or computer software to find the zeros.

Q: Can I use the quadratic formula to find the zeros of a cubic function?

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A: No, the quadratic formula is used to find the zeros of a quadratic function, not a cubic function. The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I know if a zero is a real number or a complex number?

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A: To determine if a zero is a real number or a complex number, you need to look at the factored form of the function. If the factor $(x - a)$ appears in the factored form of the function, then $a$ is a real number. If the factor $(x - a)$ appears in the factored form of the function, but $a$ is not a real number, then $a$ is a complex number.

Q: Can I use the same method to find the zeros of a polynomial function of any degree?

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A: Yes, the method of factoring and determining the multiplicities of the zeros can be used to find the zeros of a polynomial function of any degree.

Q: What if I make a mistake in finding the zeros of a cubic function?

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A: If you make a mistake in finding the zeros of a cubic function, you can try re-factoring the function or using a different method to find the zeros. You can also use a calculator or computer software to check your work.

Conclusion

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In this article, we have answered some frequently asked questions related to finding the zeros of a cubic function. We have also discussed the importance of factoring and determining the multiplicities of the zeros in finding the zeros of a cubic function.

Final Answer

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The zeros of a cubic function can be found by factoring the function and determining the multiplicities of the zeros.

Part 4 of 4: Example of { m $}$:

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Introduction

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In this article, we have discussed how to find the zeros of a cubic function and determine their multiplicities. We have also answered some frequently asked questions related to finding the zeros of a cubic function.

Example

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Let's consider the cubic function $m(x) = x^3 + 3x^2 - 36x - 108$. We can factor this function as $(x + 3)(x - 6)^2$. The zeros of this function are $x = -3$ and $x = 6$, with multiplicities 1 and 2, respectively.

Final Answer

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The zeros of the function are $x = -3, 6$ and the multiplicities are 1, 2.

Part 5 of 5: Conclusion of { m $}$:

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Introduction

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In this article, we have discussed how to find the zeros of a cubic function and determine their multiplicities. We have also answered some frequently asked questions related to finding the zeros of a cubic function.

Conclusion

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In conclusion, finding the zeros of a cubic function is an important step in understanding the behavior and properties of the function. By factoring the function and determining the multiplicities of the zeros, we can gain a deeper understanding of the function and its zeros.

Final Answer

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The zeros of a cubic function can be found by factoring the function and determining the multiplicities of the zeros.