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factoring rules

Factoring Rules

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)(a^{n– 1} + a^{n– 2}b + a^{n– 3}b² + ··· + ab^{n– 2} + b^{n– 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)(a^{n– 1} – a^{n– 2}b + a^{n– 3}b² – ··· + a²b^{n– 3} – ab^{n– 2} + b^{n– 1})
12. Sum of squares: a²+ b² = (a – bi)(a + bi) Note: a²+ b² does not factor using real numbers.
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