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) |
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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.  |