Difference between revisions of "Sum and difference of powers"
(odd, it does not work for even) |
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<math>1^3+2^3+3^3+4^3=10^2</math> | <math>1^3+2^3+3^3+4^3=10^2</math> | ||
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+ | ==Factorizations of Sums of Powers== | ||
+ | <math>x^2-y^2=(x-y)(x+y)</math> | ||
+ | |||
+ | <math>x^3-y^3=(x-y)(x^2+xy+y^2)</math> | ||
+ | |||
+ | <math>x^4-y^4=(x-y)(x^3+x^2y+xy^2+y^3)</math> | ||
+ | |||
+ | Note that all these sums of powers can be factorized as follows: | ||
+ | |||
+ | If we have a sum of powers of degree "n", then | ||
+ | |||
+ | (x^n)±(y^n)=(x±y)(x^n+(x^(n-1))y+(x^(n-2))y^2...+y^n) | ||
+ | |||
+ | Note that the "longer" polynomial in the factorization is simply a binomial theorem expansion of the binomial (x+y)^n, except for the fact that the coefficient on each of the terms is 1. This can be quite useful in problems that might have a sum of powers expression as well as an application of the binomial theorem. | ||
+ | |||
+ | - icecreamrolls8 | ||
+ | |||
==See Also== | ==See Also== | ||
* [[Factoring]] | * [[Factoring]] | ||
* [[Difference of squares]], an extremely common specific case of this. | * [[Difference of squares]], an extremely common specific case of this. | ||
+ | * [[Binomial Theorem]] | ||
{{stub}} | {{stub}} | ||
[[Category:Elementary algebra]] | [[Category:Elementary algebra]] |
Revision as of 13:12, 5 May 2021
The sum and difference of powers are powerful factoring techniques that, respectively, factor a sum or a difference of certain powers.
Contents
[hide]Sums of Odd Powers
Differences of Powers
If is a positive integer and and are real numbers,
For example:
Note that the number of terms in the long factor is equal to the exponent in the expression being factored.
An amazing thing happens when and differ by , say, . Then and
.
For example:
If we also know that then:
Sum of Cubes
Factorizations of Sums of Powers
Note that all these sums of powers can be factorized as follows:
If we have a sum of powers of degree "n", then
(x^n)±(y^n)=(x±y)(x^n+(x^(n-1))y+(x^(n-2))y^2...+y^n)
Note that the "longer" polynomial in the factorization is simply a binomial theorem expansion of the binomial (x+y)^n, except for the fact that the coefficient on each of the terms is 1. This can be quite useful in problems that might have a sum of powers expression as well as an application of the binomial theorem.
- icecreamrolls8
See Also
- Factoring
- Difference of squares, an extremely common specific case of this.
- Binomial Theorem
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