Write A Polynomial Function With The Given Zeros And Their Corresponding Multiplicities. Leave It In Factored Form.$\[ \begin{tabular}{|c|c|} \hline \text{Zeros} & \text{Multiplicity} \\ \hline 5 & 2 \\ -5 & 3 \\ -4 & 2
Introduction
In algebra, a polynomial function can be represented in various forms, including factored form. The factored form of a polynomial function is a product of linear factors, where each factor corresponds to a zero of the function. In this article, we will learn how to write a polynomial function with given zeros and their corresponding multiplicities, and leave it in factored form.
Understanding Zeros and Multiplicities
A zero of a polynomial function is a value of the variable that makes the function equal to zero. The multiplicity of a zero is the number of times the factor corresponding to that zero appears in the factored form of the polynomial function. For example, if a polynomial function has a zero at x = 5 with a multiplicity of 2, it means that the factor (x - 5) appears twice in the factored form of the polynomial function.
Given Zeros and Their Corresponding Multiplicities
In this problem, we are given the following zeros and their corresponding multiplicities:
| Zeros | Multiplicity |
|---|---|
| 5 | 2 |
| -5 | 3 |
| -4 | 2 |
Writing the Polynomial Function in Factored Form
To write the polynomial function in factored form, we need to create a factor for each zero and its corresponding multiplicity. The general form of a factor is (x - a), where a is the zero. Since the multiplicity of a zero is the number of times the factor appears, we can write the polynomial function as a product of these factors.
For the given zeros and their corresponding multiplicities, we can write the polynomial function as:
f(x) = (x - 5)^2(x + 5)^3(x + 4)^2
Explanation of the Factored Form
In the factored form of the polynomial function, each factor corresponds to a zero and its multiplicity. The factor (x - 5)^2 corresponds to the zero x = 5 with a multiplicity of 2. The factor (x + 5)^3 corresponds to the zero x = -5 with a multiplicity of 3. The factor (x + 4)^2 corresponds to the zero x = -4 with a multiplicity of 2.
Expanding the Factored Form
To expand the factored form of the polynomial function, we can use the binomial theorem. The binomial theorem states that for any positive integer n, we can expand (a + b)^n as:
(a + b)^n = ∑(n choose k)a(n-k)bk
where the sum is taken over all non-negative integers k such that 0 ≤ k ≤ n.
Using the binomial theorem, we can expand each factor in the factored form of the polynomial function:
(x - 5)^2 = x^2 - 10x + 25 (x + 5)^3 = x^3 + 15x^2 + 75x + 125 (x + 4)^2 = x^2 + 8x + 16
Multiplying the Factors
To multiply the factors, we can use the distributive property of multiplication over addition. We can multiply each term in the first factor by each term in the second factor, and so on.
Multiplying the factors, we get:
f(x) = (x^2 - 10x + 25)(x^3 + 15x^2 + 75x + 125)(x^2 + 8x + 16)
Simplifying the Expression
To simplify the expression, we can combine like terms. We can add or subtract terms with the same variable and exponent.
Simplifying the expression, we get:
f(x) = x^7 + 23x^6 + 193x^5 + 625x^4 + 1000x^3 + 725x^2 + 200x + 4000
Conclusion
In this article, we learned how to write a polynomial function with given zeros and their corresponding multiplicities, and leave it in factored form. We also learned how to expand the factored form using the binomial theorem and multiply the factors using the distributive property of multiplication over addition. Finally, we simplified the expression by combining like terms.
References
- [1] "Algebra" by Michael Artin
- [2] "Calculus" by Michael Spivak
- [3] "Polynomial Functions" by Wolfram MathWorld
Discussion
Introduction
In our previous article, we learned how to write a polynomial function with given zeros and their corresponding multiplicities, and leave it in factored form. In this article, we will answer some frequently asked questions about writing polynomial functions with given zeros and their corresponding multiplicities.
Q: What is the difference between a zero and a root of a polynomial function?
A: A zero of a polynomial function is a value of the variable that makes the function equal to zero. A root of a polynomial function is a value of the variable that makes the function equal to zero, and it is also a solution to the equation. In other words, a zero is a value that makes the function equal to zero, while a root is a value that makes the function equal to zero and is also a solution to the equation.
Q: How do I find the zeros of a polynomial function?
A: To find the zeros of a polynomial function, you can use the factored form of the function. The factored form of a polynomial function is a product of linear factors, where each factor corresponds to a zero of the function. You can also use the rational root theorem to find the zeros of a polynomial function.
Q: What is the multiplicity of a zero?
A: The multiplicity of a zero is the number of times the factor corresponding to that zero appears in the factored form of the polynomial function. For example, if a polynomial function has a zero at x = 5 with a multiplicity of 2, it means that the factor (x - 5) appears twice in the factored form of the polynomial function.
Q: How do I write a polynomial function with given zeros and their corresponding multiplicities?
A: To write a polynomial function with given zeros and their corresponding multiplicities, you can use the factored form of the function. The factored form of a polynomial function is a product of linear factors, where each factor corresponds to a zero of the function. You can also use the binomial theorem to expand the factored form of the function.
Q: Can I use the factored form of a polynomial function to solve problems in algebra and calculus?
A: Yes, you can use the factored form of a polynomial function to solve problems in algebra and calculus. The factored form of a polynomial function can be used to find the zeros of the function, which can be used to solve equations and inequalities. The factored form of a polynomial function can also be used to find the maximum and minimum values of the function.
Q: How do I expand the factored form of a polynomial function?
A: To expand the factored form of a polynomial function, you can use the binomial theorem. The binomial theorem states that for any positive integer n, we can expand (a + b)^n as:
(a + b)^n = ∑(n choose k)a(n-k)bk
where the sum is taken over all non-negative integers k such that 0 ≤ k ≤ n.
Q: Can I use the factored form of a polynomial function to find the maximum and minimum values of the function?
A: Yes, you can use the factored form of a polynomial function to find the maximum and minimum values of the function. The factored form of a polynomial function can be used to find the zeros of the function, which can be used to find the maximum and minimum values of the function.
Q: How do I simplify the expression of a polynomial function?
A: To simplify the expression of a polynomial function, you can combine like terms. You can add or subtract terms with the same variable and exponent.
Conclusion
In this article, we answered some frequently asked questions about writing polynomial functions with given zeros and their corresponding multiplicities. We also learned how to use the factored form of a polynomial function to solve problems in algebra and calculus.
References
- [1] "Algebra" by Michael Artin
- [2] "Calculus" by Michael Spivak
- [3] "Polynomial Functions" by Wolfram MathWorld
Discussion
What are some other ways to write a polynomial function with given zeros and their corresponding multiplicities? How can we use the factored form of a polynomial function to solve problems in algebra and calculus?