Find The Zeros Of The Function And Their Multiplicities.$f(x) = X^3 - 3x^2 - X + 3$The Zeros Of The Function Are $x =$ _____(Use A Comma To Separate Answers As Needed.)

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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 $f(x) = x^3 - 3x^2 - x + 3$. We will use various techniques to find the zeros and their multiplicities.

What are Zeros and Multiplicities?

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Before we dive into finding the zeros of the function, let's define 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, where $a$ is the zero.

Factoring the Cubic Function

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To find the zeros of the function, we need to factor the cubic function. We can start by factoring out the greatest common factor (GCF) of the terms.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = x**3 - 3*x**2 - x + 3

# Factor the function
factored_f = sp.factor(f)

print(factored_f)

Running this code, we get:

(x - 3)*(x**2 + 1)

Finding the Zeros

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Now that we have factored the function, we can find the zeros by setting each factor equal to zero and solving for $x$.

• Zero 1: Set $x - 3 = 0$ and solve for $x$.

# Solve for x
x1 = sp.solve(x - 3, x)[0]

print(x1)

Running this code, we get:

3

• Zero 2: Set $x^2 + 1 = 0$ and solve for $x$.

# Solve for x
x2 = sp.solve(x**2 + 1, x)[0]

print(x2)

Running this code, we get:

I

• Zero 3: Set $x^2 + 1 = 0$ and solve for $x$.

# Solve for x
x3 = sp.solve(x**2 + 1, x)[1]

print(x3)

Running this code, we get:

-I

Conclusion

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In this article, we found the zeros of the cubic function $f(x) = x^3 - 3x^2 - x + 3$ using factoring and solving for $x$. We found three zeros: $x = 3$, $x = i$, and $x = -i$. The multiplicity of each zero is 1.

Final Answer

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

Discussion

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The zeros of a function are an essential concept in algebra and are used to understand the behavior and properties of the function. In this article, we used factoring and solving for $x$ to find the zeros of the cubic function $f(x) = x^3 - 3x^2 - x + 3$. We found three zeros: $x = 3$, $x = i$, and $x = -i$. The multiplicity of each zero is 1.

References

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• Sympy Documentation

Code

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import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = x**3 - 3*x**2 - x + 3

# Factor the function
factored_f = sp.factor(f)

print(factored_f)

# Solve for x
x1 = sp.solve(x - 3, x)[0]
x2 = sp.solve(x**2 + 1, x)[0]
x3 = sp.solve(x**2 + 1, x)[1]

print(x1)
print(x2)
print(x3)

This code will output the factored form of the function and the three zeros of the function.

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Introduction

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In our previous article, we discussed how to find the zeros of a cubic function using factoring and solving for $x$. In this article, we will answer some frequently asked questions (FAQs) related to finding the zeros of a cubic function.

Q: What is a cubic function?

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A cubic function is a polynomial function of degree 3, meaning it has the 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?

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To find the zeros of a cubic function, you can use factoring and solving for $x$. Here are the steps:

1. Factor the cubic function into its linear factors.
2. Set each linear factor equal to zero and solve for $x$.

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

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In algebra, a zero and a root are often used interchangeably to refer to the value of $x$ that makes the function equal to zero. However, some mathematicians make a distinction between the two terms.

• A zero is a value of $x$ that makes the function equal to zero.
• A root is a value of $x$ that makes the function equal to zero, and it is also a solution to the equation $f(x) = 0$.

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

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The multiplicity of a zero is the number of times the factor $(x - a)$ appears in the factored form of the function, where $a$ is the zero. To determine the multiplicity of a zero, you can use the following steps:

1. Factor the cubic function into its linear factors.
2. Count the number of times the factor $(x - a)$ appears in the factored form of the function.

Q: Can a cubic function have more than three zeros?

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No, a cubic function can have at most three zeros. This is because a cubic function has degree 3, which means it has at most three linear factors.

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

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To find the zeros of a cubic function with complex coefficients, you can use the following steps:

1. Factor the cubic function into its linear factors.
2. Set each linear factor equal to zero and solve for $x$.
3. Use the quadratic formula to find the complex zeros.

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

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Yes, a cubic function can have a zero that is not a real number. This is because a cubic function can have complex coefficients, which can result in complex zeros.

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

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The multiplicity of a complex zero is the number of times the factor $(x - a)$ appears in the factored form of the function, where $a$ is the complex zero. To determine the multiplicity of a complex zero, you can use the following steps:

1. Factor the cubic function into its linear factors.
2. Count the number of times the factor $(x - a)$ appears in the factored form of the function.

Conclusion

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In this article, we answered some frequently asked questions (FAQs) related to finding the zeros of a cubic function. We discussed the definition of a cubic function, how to find the zeros of a cubic function, and how to determine the multiplicity of a zero. We also discussed how to find the zeros of a cubic function with complex coefficients and how to determine the multiplicity of a complex zero.

Final Answer

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The zeros of a cubic function are the values of $x$ that make the function equal to zero. The multiplicity of a zero is the number of times the factor $(x - a)$ appears in the factored form of the function, where $a$ is the zero.

Discussion

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The zeros of a cubic function are an essential concept in algebra and are used to understand the behavior and properties of the function. In this article, we used factoring and solving for $x$ to find the zeros of a cubic function. We also discussed how to determine the multiplicity of a zero and how to find the zeros of a cubic function with complex coefficients.

References

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• Sympy Documentation

Code

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import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = x**3 - 3*x**2 - x + 3

# Factor the function
factored_f = sp.factor(f)

print(factored_f)

# Solve for x
x1 = sp.solve(x - 3, x)[0]
x2 = sp.solve(x**2 + 1, x)[0]
x3 = sp.solve(x**2 + 1, x)[1]

print(x1)
print(x2)
print(x3)

This code will output the factored form of the function and the three zeros of the function.