Factorization example, Mathematics

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Example 

Factorize x2 - 4x + 4.

If we substitute x = 1, the value of the expression will be (1)2 - 4(1) + 4 = 1

If we substitute x = -1, the value of the expression will be (-1)2 - 4(-1) +  4 = 9

If we substitute x = 2, the value of the expression will be (2)2 - 4(2) + 4  = 0

For x = 2, the value of the expression is 0. That is, x - 2 (observe that x - 2 = 0 and x = 2 are one and the same) is one of the factors of the expression x2 - 4x + 4. To obtain the other factor we divide the expression by the factor we got. That will be

x - 2 )

x-  4x + 4

( x - 2

       (-)

x2  -  2x

 


 

   -  2x + 4

 

       (-)

       -  2x + 4

 


 

                      0

 


From the division we observe that x - 2 is the other factor. When this is equated to zero we obtain x = 2. Therefore, the factors of x2 - 4x + 4 are (x - 2)(x - 2) or (x - 2)2.

Now, we look at another identity which is similar to the one you have seen earlier except the (-) sign. The identity is (a - b)2 = a2 - 2ab + b2. The advantage of being familiar with identities is that you do not have to sweat it out by factorizing each and every expression you are given. On the other hand it is not mandatory that each and every expression given should be in conformation with some identity. In this case there is no easy way out except solving the problem by trial and error method to start with and then go for division in order to know other factors.

Another identity of second degree we often come across is 

a2 - b2 = (a + b)(a - b)

According to this identity the difference of squares of any two quantities is equal to the product of the sum and the difference of the two quantities.


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