Special Products and Factors

Special Products and Factors

1. $(x+y)^2 =x^2+2xy+y^2$

2. $(x-y)^2=x^2-2xy+y^2$

3. $(x+y)^3=x^3+3x^2y+3xy^2+y^3$

4. $(x-y)^3=x^3-3x^2y+3xy^2-y^3$

5. $(x+y)^4 = x^4+4x^3y+6x^2y^2+4xy^3+y^4$

6. $(x-y)^4= x^4-4x^3y+6x^2y^2-4xy^3+y^4$

7. $(x+y)^5=x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5$

8. $(x-y)^5=x^5-5x^4y+10x^3y^2+10x^3y^2-10x^2y^3+5xy^4-y^5$

9. $(x+y)^6=x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6$

10. $(x-y)^6=x^6-6x^5y+15x^4y^2-20x^3y^3+15x^2y^4-6xy^5+y^6$

The results 1 to 10 above are special cases of the $binomial formula$

11. $x^2-y^2=(x-y)(x+y)$

12. $x^3-y^3=(x-y)(x^2+xy+y^2)$

13. $x^3+y^3 =(x+y)(x^2-xy+y^2)$

14. $x^4-y^4=(x-y)(x+y)(x^2+y^2)$

15. $x^5-y^5=(x-y)(x^4+x^3y+x^2y^2+xy^3+y^4)$

16. $x^5+y^5=(x+y)(x^4-x^3y+x^2y^2-xy^3+y^4)$

17. $x^6-y^6=(x-y)(x+y)(x^2+xy+y^2)(x^2-xy+y^2)$

18. $x^4+x^2y^2+y^4=(x^2+xy+y^2)(x^2-xy+y^2)$

19. $x^4+4y^4 =(x^2+2xy+2y^2)(x^2-2xy+2y^2)$

Some generalization of the above are given by the following results where $n$ is a position integer .

20. $x^{2n+1}-y^{2n+1}=(x-y)(x^{2n}+x^{2n-1}y+x^{2n-2}y^2+\dots +y^{2n})$

                                          $= (x-y)(x^2-2xy\cos\cfrac{2\pi}{2n+1}+y^2)(x^2-2xy\cos\cfrac{4\pi}{2n+1}+y^2)\dots (x^2-2xy\cos\cfrac{2n\pi}{2n+1}+y^2)$

21. $x^{2n+1}+y^{2n+1}=(x+y)(x^{2n}-x^{2n-1}y+x^{2n-2}y^2-\dots +y^{2n})$
                                           $=(x+y)(x^2+2xy\cos\cfrac{2\pi}{2n+1}+y^2)(x^2+2xy\cos\cfrac{4\pi}{2n+1}+y^2)\dots (x^2+2xy\cos\cfrac{2n\pi}{2n+1}+y^2)$

22. $x^{2n}-y^{2n}=(x-y)(x+y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+\dots)(x^{n-1}-x^{n-2}y+x^{n-3}y^2-\dots)$
                               $=(x-y)(x+y)(x^2-2xy\cos\cfrac{\pi}{n}+y^2)(x^2-2xy\cos\cfrac{2\pi}{n}+y^2)$
                                $\dots (x^2-2xy\cos\cfrac{(n-1)\pi}{n}+y^2)$

23. $x^{2n}+y^{2n}=(x^2+2xy\cos\cfrac{\pi}{2n}+y^2)(x^2+2xy\cos\cfrac{3\pi}{2n}+y^2)$
                                  $\dots (x^2+2xy\cos\cfrac{(2n-1)\pi}{2n}+y^2)$