Factoring Polynomials

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Solution of a simple equation depends only on the following axioms:-
 
1. If to equal we add equals the sums are equal.
2. If from equals we take equals the remainders are equal.
3. If equals are multiplied by equals the products are equal.
4. If equals are divided by equals the quotients are equal.
 
Example 1.To solve the equation 7 x = 14 dividing both sides by 7 we get
                   x = 2  
 
Example 2. To solve the equation x/2 = -6 multiplying by 2 both sides we get
                     x = - 12  
 
Example 3. To solve the equation 7x – 2 x – x = 10 – 23 –15
                    By collecting terms on each side, we get
                    4 x = - 28
                    dividing by 4, we get  x = - 7
 
Example 4. Resolve into factors 2 x2y3 – 6 x2y2 + 2 xy3
                    2 x2y3 – 6 x2y2 + 2 xy3
                   = 2 xy (xy2 – 3 xy + y2)
                   = 2 xy2 (xy – 3 x + y)
 
Example 5. Resolve into factors 6 x3 – 9 x2y + 12 x y2
                           6 x3 – 9 x2y + 12 x y
                          = 3 x (2 x – 3 xy + 4 y2 )

Example 6. Resolve the factors  x2 – ax + bx - ab      
                   x2 – ax + bx - ab  = (x2 – ax) + (bx – ab)
                   x2 – ax + bx – ab =  x (x - a) + b (x – a)
                   x2 – ax + bx – ab   =  (x - a)  (x + b)
 

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