Sum and difference of powers
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 difference of powers of degree , then
where a, b, c,... are equivalent to n-1, n-2, n-3,..., respectively.
Note, of course, that some of the signs simply change when we have sum of powers instead of difference. Regardless, observe 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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