Factoring Rules

Algebra formulas for factoring.

1.   x2 – (r + s)x + rs = (x – r)(x – s)
2.   x2 + 2ax + a2 = (x + a)2  and  x2 – 2ax + a2 = (x – a)2
3.   Difference of squares:  a2 – b2 = (a – b)(a + b)
4.   Difference of cubes:  a3 – b3 = (a – b)(a2 + ab + b2)
5.   a4 – b4 = (a – b)(a3 + a2b + ab2 + b3) = (a – b) [ a2(a + b) + b2(a + b) ] = (a – b)(a + b)(a2 + b2)
	or, more simply,   a4 – b4 = (a2 – b2)(a2 + b2) = (a – b)(a + b)(a2 + b2)
6.   a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
7.   an – bn = (a – b)(an – 1 + an – 2b + an – 3b2 + ··· + abn – 2 + bn – 1)
8.   Sum of cubes:  a3 + b3 = (a + b)(a2 – ab + b2)
9.   a5 + b5 = (a + b)(a4 – a3b + a2b2 – ab3 + b4)
10.   a7 + b7 = (a + b)(a6 – a5b + a4b2 – a3b3 + a2b4 – ab5 + b6)
11.   If n is odd, then  an + bn = (a + b)(an – 1 – an – 2b + an – 3b2 – ··· + a2bn – 3 – abn – 2 + bn – 1)
12.   Sum of squares:  a2 + b2 = (a – bi)(a + bi)      Note:  a2 + b2  does not factor using real numbers.
13.

See also

Factor theorem, rational root theorem, polynomial long division, synthetic division