Factoring
Factoring is an essential part of problem solving. It is part of number theory and applying it will yield many results. There are many methods to factor, ranging all the way from using the Distributive Property to make an expression easier to calculate to using factoring as a vital part of proofs.
Contents
Differences and Sums of Powers
Using the formula for the sum of a geometric sequence, it's easy to derive the more general formula:
Take note of the specific case where n is odd:
This also leads to the formula for the sum of cubes,
Vieta's/Newton Factorizations
These factorizations are useful for problems that could otherwise be solved by Newton sums or problems that give a polynomial and ask a question about the roots. Combined with Vieta's formulas, these are excellent factorizations that show up everywhere.
Other Useful Factorizations
- See Simon's Favorite Factoring Trick (This is not a recognized formula, please do not quote it on contests)
- Binomial theorem
Practice Problems
- Prove that
is never divisible by 121 for any positive integer
.
- Prove that
is divisible by 7. - USSR Problem Book
- Factor
.