Difference between revisions of "Sum and difference of powers"
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Note that all these sums of powers can be factorized as follows: | Note that all these sums of powers can be factorized as follows: | ||
− | If we have a sum of powers of degree | + | If we have a sum of powers of degree <math>n</math>, then |
− | + | <cmath>x^n±y^n=(x±y)(x^n+x^n-1y+x^n-2y^2...+y^n)</cmath> | |
− | 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. | + | Note that the "longer" polynomial in the factorization is simply a binomial theorem expansion of the binomial <math>(x+y)^n</math>, except for the fact that the coefficient on each of the terms is <math>1</math>. 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 | - icecreamrolls8 |
Revision as of 13:54, 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 , then
Note that the "longer" polynomial in the factorization is simply a binomial theorem expansion of the binomial , except for the fact that the coefficient on each of the terms is . 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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