Write A Polynomial Function In Factored And Standard Form With The Given Zeros: $x=0, 5, 2$.Factored Form: $f(x)=(x)(x-5)(x-2$\]Standard Form: $f(x)=$ $\square$
Introduction
In algebra, polynomial functions are a crucial concept in mathematics, and understanding how to write them in factored and standard form is essential for solving various mathematical problems. In this article, we will explore how to write a polynomial function in factored and standard form given the zeros of the function.
What are Zeros in a Polynomial Function?
The zeros of a polynomial function are the values of x that make the function equal to zero. In other words, they are the x-intercepts of the graph of the function. For example, if we have a polynomial function f(x) = (x)(x-5)(x-2), the zeros of the function are x = 0, x = 5, and x = 2.
Factored Form of a Polynomial Function
The factored form of a polynomial function is a product of linear factors, where each factor corresponds to a zero of the function. In the case of the given zeros x = 0, 5, and 2, the factored form of the polynomial function is:
f(x) = (x)(x-5)(x-2)
This form is called the factored form because it is a product of factors, each of which corresponds to a zero of the function.
Standard Form of a Polynomial Function
The standard form of a polynomial function is a sum of terms, where each term is a power of x. To convert the factored form of a polynomial function to standard form, we need to multiply out the factors. In the case of the given factored form f(x) = (x)(x-5)(x-2), we can multiply out the factors as follows:
f(x) = (x)(x-5)(x-2) = x(x^2 - 7x + 10) = x^3 - 7x^2 + 10x
Therefore, the standard form of the polynomial function is f(x) = x^3 - 7x^2 + 10x.
Why is it Important to Write Polynomial Functions in Both Factored and Standard Form?
Writing polynomial functions in both factored and standard form is important for several reasons:
- Simplifying Complex Functions: Factored form can be used to simplify complex functions by breaking them down into simpler factors.
- Finding Zeros: Factored form can be used to find the zeros of a function by setting each factor equal to zero and solving for x.
- Graphing Functions: Standard form can be used to graph functions by plotting the x-intercepts and other key features of the graph.
- Solving Equations: Standard form can be used to solve equations by using algebraic techniques such as factoring and the quadratic formula.
Conclusion
In conclusion, writing polynomial functions in factored and standard form is an essential skill in mathematics. By understanding how to write functions in both forms, we can simplify complex functions, find zeros, graph functions, and solve equations. In this article, we have explored how to write a polynomial function in factored and standard form given the zeros of the function.
Examples and Exercises
Here are some examples and exercises to help you practice writing polynomial functions in factored and standard form:
Example 1
Write the polynomial function in factored form given the zeros x = 1, 2, and 3.
f(x) = (x-1)(x-2)(x-3)
Example 2
Write the polynomial function in standard form given the factored form f(x) = (x-1)(x-2)(x-3).
f(x) = x^3 - 6x^2 + 11x - 6
Exercise 1
Write the polynomial function in factored form given the zeros x = 0, 4, and 6.
Exercise 2
Write the polynomial function in standard form given the factored form f(x) = (x)(x-4)(x-6).
Exercise 3
Find the zeros of the polynomial function f(x) = x^3 - 6x^2 + 11x - 6.
Exercise 4
Graph the polynomial function f(x) = x^3 - 6x^2 + 11x - 6.
Glossary of Terms
Here are some key terms related to polynomial functions:
- Zeros: The values of x that make the function equal to zero.
- Factored Form: A product of linear factors, where each factor corresponds to a zero of the function.
- Standard Form: A sum of terms, where each term is a power of x.
- Polynomial Function: 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 ≠0 and n is a non-negative integer.
References
Here are some references for further reading on polynomial functions:
- Algebra: A comprehensive textbook on algebra that covers polynomial functions in detail.
- Calculus: A textbook on calculus that covers polynomial functions in the context of limits and derivatives.
- Mathematics Online Resources: A website that provides online resources and tutorials on polynomial functions and other mathematical topics.
Polynomial Functions Q&A ==========================
Frequently Asked Questions
Here are some frequently asked questions about polynomial functions:
Q1: What is a polynomial function?
A1: 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 ≠0 and n is a non-negative integer.
Q2: What is the difference between factored and standard form of a polynomial function?
A2: The factored form of a polynomial function is a product of linear factors, where each factor corresponds to a zero of the function. The standard form of a polynomial function is a sum of terms, where each term is a power of x.
Q3: How do I find the zeros of a polynomial function?
A3: To find the zeros of a polynomial function, you can set each factor equal to zero and solve for x. Alternatively, you can use the quadratic formula to find the zeros of a quadratic function.
Q4: How do I graph a polynomial function?
A4: To graph a polynomial function, you can plot the x-intercepts and other key features of the graph. You can also use a graphing calculator or software to graph the function.
Q5: What is the degree of a polynomial function?
A5: The degree of a polynomial function is the highest power of x in the function. For example, the degree of the polynomial function f(x) = x^3 + 2x^2 + 3x + 1 is 3.
Q6: How do I add or subtract polynomial functions?
A6: To add or subtract polynomial functions, you can combine like terms. For example, to add the polynomial functions f(x) = x^2 + 2x + 1 and g(x) = x^2 + 3x + 2, you can combine like terms to get f(x) + g(x) = 2x^2 + 5x + 3.
Q7: How do I multiply polynomial functions?
A7: To multiply polynomial functions, you can use the distributive property to multiply each term of one function by each term of the other function. For example, to multiply the polynomial functions f(x) = x^2 + 2x + 1 and g(x) = x^2 + 3x + 2, you can use the distributive property to get f(x)g(x) = x^4 + 5x^3 + 10x^2 + 8x + 2.
Q8: What is the leading coefficient of a polynomial function?
A8: The leading coefficient of a polynomial function is the coefficient of the highest power of x in the function. For example, the leading coefficient of the polynomial function f(x) = x^3 + 2x^2 + 3x + 1 is 1.
Q9: How do I divide polynomial functions?
A9: To divide polynomial functions, you can use long division or synthetic division. For example, to divide the polynomial functions f(x) = x^3 + 2x^2 + 3x + 1 by g(x) = x + 1, you can use long division to get f(x)/g(x) = x^2 + x + 1.
Q10: What is the remainder of a polynomial function?
A10: The remainder of a polynomial function is the amount left over after dividing the function by another function. For example, the remainder of the polynomial function f(x) = x^3 + 2x^2 + 3x + 1 divided by g(x) = x + 1 is 0.
Additional Resources
Here are some additional resources for learning more about polynomial functions:
- Algebra: A comprehensive textbook on algebra that covers polynomial functions in detail.
- Calculus: A textbook on calculus that covers polynomial functions in the context of limits and derivatives.
- Mathematics Online Resources: A website that provides online resources and tutorials on polynomial functions and other mathematical topics.
Glossary of Terms
Here are some key terms related to polynomial functions:
- Zeros: The values of x that make the function equal to zero.
- Factored Form: A product of linear factors, where each factor corresponds to a zero of the function.
- Standard Form: A sum of terms, where each term is a power of x.
- Polynomial Function: 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 ≠0 and n is a non-negative integer.
- Degree: The highest power of x in the function.
- Leading Coefficient: The coefficient of the highest power of x in the function.
- Remainder: The amount left over after dividing the function by another function.