Rucete ✏ Mathematics In a Nutshell
Factorization | In a nutshell
Factorization is the process of expressing a number or any mathematical object as the product of several factors, which are typically smaller or simpler objects of the same type.
Rule
We can obtain factors by reversing the previously taught expansion.
In particular, for factorization, memorizing formulas is much more important than expansion because it makes the problem much easier to solve.
1. xa + xb + xc = x(a + b + c)
2. a² + 2ab + b² = (a + b)²
3. a² - 2ab + b² = (a - b)²
4. a² - b² = (a + b)(a - b)
5. x² + (a + b)x + ab = (x + a)(x - b)
6. a² + b² + c² + 2(ab + bc + ca) = (a + b + c)²
7. acx² + (ad + bc)x + bd = (ax + b)(cx + d)
8. a³ ± 3a²b + 3ab² ± b³ = (a ± b)³
9. a³ + b³ = (a + b)(a² - ab + b²)
10. a³ - b³ = (a - b)(a² + ab + b²)
11. a³ + b³ + c³ - 3abc =(a + b + c)(a² + b² + c² - ab - bc - ca) or 1/2(a + b + c){(a - b)² + (b - c)² + (c - a)²}
Example
(1) x³y - xy³
=xy(x + y)(x - y)
(2) sin² x + 2sin x + 1
=(sin x +1)²
(3) 49 - x²
= (7 + x)(7 - x)
(4) a(x + 5) - b(x + 5)
= (x + 5)(a - b)
(5) 2x² -15x + 7
=(2x - 1)(x - 7)
(6) x³ - 3x²y + 3xy² - y³
= (x - y)³