Write An Equation For The Cubic Polynomial Function Whose Graph Has Zeroes At 2, 3, And 5.Start With The Polynomial Function:$\[ F(x) = (x - 2)(x - 3)(x - 5) \\]Simplify The Right Side. What Is The Equation?A. $\[ F(x) = X^3 + 31x - 30

Introduction

In algebra, a cubic polynomial function is a polynomial function of degree three, which means the highest power of the variable is three. The general form of a cubic polynomial function is $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants. In this article, we will focus on finding the equation of a cubic polynomial function whose graph has zeroes at 2, 3, and 5.

Given Polynomial Function

The given polynomial function is $f(x) = (x - 2)(x - 3)(x - 5)$. This function has zeroes at 2, 3, and 5, which means that the graph of the function will touch the x-axis at these points.

Simplifying the Right Side

To simplify the right side of the given function, we can use the distributive property to multiply the three binomials. This will give us a polynomial function in the form of $f(x) = ax^3 + bx^2 + cx + d$.

Step 1: Multiply the First Two Binomials

First, we will multiply the first two binomials: $(x - 2)$ and $(x - 3)$. This will give us:

$(x - 2)(x - 3) = x^2 - 3x - 2x + 6$

Step 2: Simplify the Expression

Now, we will simplify the expression by combining like terms:

$x^2 - 3x - 2x + 6 = x^2 - 5x + 6$

Step 3: Multiply the Result by the Third Binomial

Next, we will multiply the result by the third binomial: $(x - 5)$. This will give us:

$(x^2 - 5x + 6)(x - 5) = x^3 - 5x^2 + 6x - 5x^2 + 25x - 30$

Step 4: Simplify the Expression

Now, we will simplify the expression by combining like terms:

$x^3 - 5x^2 + 6x - 5x^2 + 25x - 30 = x^3 - 10x^2 + 31x - 30$

The Final Equation

Therefore, the equation of the cubic polynomial function whose graph has zeroes at 2, 3, and 5 is:

$f(x) = x^3 - 10x^2 + 31x - 30$

Conclusion

In this article, we have found the equation of a cubic polynomial function whose graph has zeroes at 2, 3, and 5. The equation is $f(x) = x^3 - 10x^2 + 31x - 30$. This equation can be used to model real-world situations where a cubic relationship exists.

Discussion

The equation of a cubic polynomial function can be used to model a wide range of real-world situations, including population growth, chemical reactions, and electrical circuits. In each of these situations, the equation can be used to predict the behavior of the system over time.

Example

For example, suppose we want to model the population of a city over time. We can use the equation $f(x) = x^3 - 10x^2 + 31x - 30$ to predict the population at different times. By plugging in different values of x, we can see how the population changes over time.

Applications

The equation of a cubic polynomial function has many applications in science and engineering. For example, it can be used to model the motion of a projectile, the flow of a fluid, and the vibration of a spring.

Limitations

However, the equation of a cubic polynomial function also has some limitations. For example, it can only be used to model situations where the relationship between the variables is cubic. If the relationship is not cubic, then the equation will not be accurate.

Future Research

In the future, researchers may want to explore the use of cubic polynomial functions in new and innovative ways. For example, they may want to use the equation to model complex systems that involve multiple variables and interactions.

Conclusion

In conclusion, the equation of a cubic polynomial function whose graph has zeroes at 2, 3, and 5 is $f(x) = x^3 - 10x^2 + 31x - 30$. This equation can be used to model real-world situations where a cubic relationship exists, and it has many applications in science and engineering. However, it also has some limitations, and researchers may want to explore new and innovative ways to use the equation in the future.

Introduction

In our previous article, we discussed the cubic polynomial function and how to find its equation when given the zeroes. In this article, we will answer some frequently asked questions about cubic polynomial functions.

Q: What is a cubic polynomial function?

A: A cubic polynomial function is a polynomial function of degree three, which means the highest power of the variable is three. The general form of a cubic polynomial function is $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants.

Q: How do I find the equation of a cubic polynomial function when given the zeroes?

A: To find the equation of a cubic polynomial function when given the zeroes, you can use the factored form of the function: $f(x) = (x - r_1)(x - r_2)(x - r_3)$, where $r_1$, $r_2$, and $r_3$ are the zeroes. You can then multiply out the factors to get the equation in the form $f(x) = ax^3 + bx^2 + cx + d$.

Q: What is the difference between a cubic polynomial function and a quadratic polynomial function?

A: A quadratic polynomial function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic polynomial function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. A cubic polynomial function, on the other hand, is a polynomial function of degree three, which means the highest power of the variable is three.

Q: Can I use a cubic polynomial function to model a real-world situation?

A: Yes, you can use a cubic polynomial function to model a real-world situation. For example, you can use it to model the population of a city over time, the flow of a fluid, or the vibration of a spring.

Q: What are some common applications of cubic polynomial functions?

A: Some common applications of cubic polynomial functions include:

• Modeling population growth
• Modeling chemical reactions
• Modeling electrical circuits
• Modeling the motion of a projectile
• Modeling the flow of a fluid
• Modeling the vibration of a spring

Q: What are some limitations of cubic polynomial functions?

A: Some limitations of cubic polynomial functions include:

• They can only be used to model situations where the relationship between the variables is cubic.
• They may not be accurate for modeling situations where the relationship between the variables is not cubic.
• They may not be able to capture complex behavior in a system.

Q: Can I use a cubic polynomial function to model a system with multiple variables and interactions?

A: No, a cubic polynomial function is not suitable for modeling a system with multiple variables and interactions. In such cases, you may need to use a more complex mathematical model, such as a system of differential equations.

Q: How do I determine the degree of a polynomial function?

A: To determine the degree of a polynomial function, you can look at the highest power of the variable. For example, if the polynomial function is $f(x) = ax^3 + bx^2 + cx + d$, then the degree of the function is three.

Q: Can I use a cubic polynomial function to model a system with a non-linear relationship between the variables?

A: Yes, you can use a cubic polynomial function to model a system with a non-linear relationship between the variables. However, you may need to use a more complex mathematical model, such as a system of differential equations, to capture the complex behavior in the system.

Conclusion

In this article, we have answered some frequently asked questions about cubic polynomial functions. We have discussed the definition of a cubic polynomial function, how to find its equation when given the zeroes, and some common applications and limitations of cubic polynomial functions. We have also discussed how to determine the degree of a polynomial function and how to use a cubic polynomial function to model a system with a non-linear relationship between the variables.