A Polynomial Function Has A Zero At -6 With Multiplicity 2 And A Zero At 2 With Multiplicity 1. Write A Polynomial Function In Standard Form That Could Represent This Function.

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Understanding the Problem

When dealing with polynomial functions, it's essential to understand the concept of zeros and their multiplicities. A zero of a polynomial function is a value of the variable (in this case, x) that makes the function equal to zero. The multiplicity of a zero refers to the number of times the factor corresponding to that zero appears in the polynomial's factored form.

In this problem, we are given two zeros: -6 with a multiplicity of 2 and 2 with a multiplicity of 1. This means that the polynomial function will have two factors: (x + 6)^2 and (x - 2).

Writing the Polynomial Function

To write the polynomial function in standard form, we need to multiply the factors together and then expand the result. The standard form of a polynomial function is a sum of terms, where each term is a constant multiplied by a power of x.

Let's start by multiplying the factors:

(x + 6)^2(x - 2)

Expanding this expression, we get:

(x^2 + 12x + 36)(x - 2)

Now, we need to multiply the two binomials together:

x^2(x - 2) + 12x(x - 2) + 36(x - 2)

Expanding each term, we get:

x^3 - 2x^2 + 12x^2 - 24x + 36x - 72

Combining like terms, we get:

x^3 + 10x^2 + 12x - 72

Simplifying the Polynomial Function

The polynomial function we obtained is in the form of a cubic polynomial. However, we can simplify it further by factoring out any common factors.

In this case, we can factor out a 2 from the constant term:

x^3 + 10x^2 + 12x - 36(2)

This simplifies the polynomial function to:

x^3 + 10x^2 + 12x - 72

Conclusion

In this article, we have written a polynomial function in standard form that represents the given zeros and their multiplicities. The polynomial function is:

x^3 + 10x^2 + 12x - 72

This function has a zero at -6 with multiplicity 2 and a zero at 2 with multiplicity 1.

Understanding the Graph of the Polynomial Function

The graph of a polynomial function can provide valuable information about its behavior and characteristics. In this case, the graph of the polynomial function will have two x-intercepts: -6 and 2.

The x-intercept at -6 will be a double root, meaning that the graph will touch the x-axis at this point and then turn back up. The x-intercept at 2 will be a single root, meaning that the graph will cross the x-axis at this point.

Using the Polynomial Function in Real-World Applications

Polynomial functions have many real-world applications, including modeling population growth, chemical reactions, and electrical circuits. In each of these cases, the polynomial function can be used to make predictions and analyze data.

For example, if we were modeling the growth of a population, we could use a polynomial function to predict the population size at different times. If we were analyzing the behavior of an electrical circuit, we could use a polynomial function to model the circuit's behavior and make predictions about its performance.

Conclusion

In this article, we have written a polynomial function in standard form that represents the given zeros and their multiplicities. The polynomial function is:

x^3 + 10x^2 + 12x - 72

This function has a zero at -6 with multiplicity 2 and a zero at 2 with multiplicity 1. We have also discussed the graph of the polynomial function and its real-world applications.

Frequently Asked Questions

  • Q: What is the multiplicity of a zero in a polynomial function? A: The multiplicity of a zero is the number of times the factor corresponding to that zero appears in the polynomial's factored form.
  • Q: How do you write a polynomial function in standard form? A: To write a polynomial function in standard form, you need to multiply the factors together and then expand the result.
  • Q: What is the graph of a polynomial function? A: The graph of a polynomial function is a curve that shows the relationship between the variable (x) and the function's value.

References

  • [1] "Polynomial Functions" by Math Open Reference
  • [2] "Graphing Polynomial Functions" by Khan Academy
  • [3] "Real-World Applications of Polynomial Functions" by Wolfram Alpha

Further Reading

  • "Polynomial Functions: A Comprehensive Guide" by Springer
  • "Graphing Polynomial Functions: A Step-by-Step Guide" by CRC Press
  • "Real-World Applications of Polynomial Functions: A Case Study" by IEEE Xplore

Q&A Article

In this article, we will answer some of the most frequently asked questions about polynomial functions. Whether you are a student, a teacher, or a professional, this article will provide you with a comprehensive understanding of polynomial functions and their applications.

Q: What is a polynomial function?

A: A polynomial function is a function that can be written in the form:

f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n, a_(n-1), ..., a_1, a_0 are constants, and n is a non-negative integer.

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

A: A polynomial function is a function that can be written in the form of a polynomial, whereas a rational function is a function that can be written in the form of a ratio of two polynomials.

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

A: The degree of a polynomial function is the highest power of the variable (x) in the polynomial. For example, the degree of the polynomial function f(x) = x^3 + 2x^2 + 3x + 1 is 3.

Q: What is the leading coefficient of a polynomial function?

A: The leading coefficient of a polynomial function is the coefficient of the highest power of the variable (x) in the polynomial. For example, the leading coefficient of the polynomial function f(x) = x^3 + 2x^2 + 3x + 1 is 1.

Q: How do you factor a polynomial function?

A: Factoring a polynomial function involves expressing it as a product of simpler polynomials. There are several methods for factoring polynomial functions, including:

  • Factoring out the greatest common factor (GCF)
  • Factoring by grouping
  • Factoring using the difference of squares formula
  • Factoring using the sum and difference of cubes formula

Q: What is the graph of a polynomial function?

A: The graph of a polynomial function is a curve that shows the relationship between the variable (x) and the function's value. The graph of a polynomial function can be a straight line, a parabola, a cubic curve, or a higher-degree curve.

Q: How do you find the x-intercepts of a polynomial function?

A: The x-intercepts of a polynomial function are the values of x that make the function equal to zero. To find the x-intercepts of a polynomial function, you can set the function equal to zero and solve for x.

Q: How do you find the y-intercept of a polynomial function?

A: The y-intercept of a polynomial function is the value of the function when x is equal to zero. To find the y-intercept of a polynomial function, you can substitute x = 0 into the function and evaluate it.

Q: What is the domain of a polynomial function?

A: The domain of a polynomial function is the set of all possible input values (x) for which the function is defined. The domain of a polynomial function is typically all real numbers.

Q: What is the range of a polynomial function?

A: The range of a polynomial function is the set of all possible output values (y) for which the function is defined. The range of a polynomial function can be all real numbers, a subset of real numbers, or a single value.

Q: How do you determine if a polynomial function is increasing or decreasing?

A: To determine if a polynomial function is increasing or decreasing, you can examine the derivative of the function. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.

Q: What is the derivative of a polynomial function?

A: The derivative of a polynomial function is a new function that represents the rate of change of the original function. The derivative of a polynomial function can be found using the power rule of differentiation.

Q: How do you use polynomial functions in real-world applications?

A: Polynomial functions have many real-world applications, including:

  • Modeling population growth
  • Analyzing chemical reactions
  • Studying electrical circuits
  • Predicting stock prices
  • Optimizing business decisions

Conclusion

In this article, we have answered some of the most frequently asked questions about polynomial functions. Whether you are a student, a teacher, or a professional, this article will provide you with a comprehensive understanding of polynomial functions and their applications.

Frequently Asked Questions

  • Q: What is the difference between a polynomial function and a rational function? A: A polynomial function is a function that can be written in the form of a polynomial, whereas a rational function is a function that can be written in the form of a ratio of two polynomials.
  • Q: How do you determine the degree of a polynomial function? A: The degree of a polynomial function is the highest power of the variable (x) in the polynomial.
  • Q: What is the leading coefficient of a polynomial function? A: The leading coefficient of a polynomial function is the coefficient of the highest power of the variable (x) in the polynomial.

References

  • [1] "Polynomial Functions" by Math Open Reference
  • [2] "Graphing Polynomial Functions" by Khan Academy
  • [3] "Real-World Applications of Polynomial Functions" by Wolfram Alpha

Further Reading

  • "Polynomial Functions: A Comprehensive Guide" by Springer
  • "Graphing Polynomial Functions: A Step-by-Step Guide" by CRC Press
  • "Real-World Applications of Polynomial Functions: A Case Study" by IEEE Xplore