Draft:Factorization of sums and differences of powers
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Submission declined on 12 December 2023 by MicrobiologyMarcus (talk). This draft's references do not show that the subject qualifies for a Wikipedia article. In summary, the draft needs multiple published sources that are: Declined by MicrobiologyMarcus 5 months ago.
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- Comment: This content is already covered at Factorization#Recognizable patterns. So, one can use this title for a redirect to this section. Otherwise, creating an article with this title would be a WP:REDUNDANTFORK
- Comment: A redirect has already been created. Unless someone objects in the next few days I will delete
- Comment: This technique needs to demonstrate notability by showing coverage in secondary sources. Further, the derivations need to be sourced so as to not appear to be original research. microbiologyMarcus (petri dish·growths) 17:36, 12 December 2023 (UTC)
- Comment: I think this idea that this is a duplicate makes sense from the point of view of a mathematician but not from the more enlightened and intelligent point of view of a teacher of mathematics. Michael Hardy (talk) 04:17, 28 December 2023 (UTC)
- Comment: This content is already covered at Factorization#Recognizable patterns. So, one can use this title for a redirect to this section. Otherwise, creating an article with this title would be a WP:REDUNDANTFORK.
- Comment: A redirect has already been created. Unless someone objects in the next few days I will delete— Preceding unsigned comment added by Ldm1954 (talk • contribs)
In mathematics, sums and differences of powers are expressions of the form and , respectively, where is a positive integer. They can be factored by a method that is a generalization of the factorization of the difference of squares and the sum of cubes.
Difference of powers[edit]
The expression can be factored by a method concretely exemplified by this case:
That pattern works with other powers than the 5th, as follows:[1]
where the second one is written in sigma notation.
Table of special cases[edit]
The first five cases are as follows:
n | expression: | factorization |
---|---|---|
1 | ||
2 | ||
3 | ||
4 | ||
5 |
Proof[edit]
To show that the factorization is true, expand the expression on the right-hand side:[2][1]
Use in calculus[edit]
Early in a beginning calculus course, one learns the power rule:
One way to prove this uses the factorization of powers, illustrated here in the case
Sum of odd powers[edit]
The expression can be factored as follows:[3]
Table of values[edit]
Table of values for :
n | expression: | factorization |
---|---|---|
1 | ||
2 | ||
3 | ||
4 | ||
5 |
Proof[edit]
The sum of powers factorization can be derived from the difference of powers factorization.[4][5]
Let and substitute this expression into the previous equation.
An negative number raised to an odd exponent will result in a negative number. Similarly, if a negative number is raised to an even exponent, it will result in a positive number. Using such reasoning, one can deduct that .
Complex conjugate[edit]
Given that , can be factored using [[complex numbers]s.
Using this property, all expressions in the form can factored as .
Examples[edit]
Example 1[edit]
Factorization of .
Example 2[edit]
Factorization of .
References[edit]
- ^ a b "Difference of Two Powers – ProofWiki". proofwiki.org. Retrieved 2023-12-12.
- ^ Andrews, George E. (1994-10-12). Number Theory. Courier Corporation. ISBN 978-0-486-68252-5.
- ^ Axler, Sheldon Jay (2013). Precalculus: a prelude to calculus (3rd ed.). Hoboken: Wiley. ISBN 978-0-470-64804-9.
- ^ Spiegel, Murray; Lipschutz, Seymour; Liu, John (2008-08-31). Schaum's Outline of Mathematical Handbook of Formulas and Tables, 3ed. McGraw Hill Professional. ISBN 978-0-07-154856-4.
- ^ "Sum of Two Odd Powers – ProofWiki". proofwiki.org. Retrieved 2023-12-12.
- ^ Gallian, Joseph A. (2021). Contemporary abstract algebra. Textbooks in mathematics (10th ed.). Boca Raton (Fla.): CRC press. ISBN 978-1-000-33735-8.
External links[edit]
- Sum and difference of powers at Art of Problem Solving
- Difference of powers at themathpage.com