x^3+y^3

Factorization of x³ + y³

                  It can be seen in most book that x³ + y³ can be factorized by dividing the expression by  (x + y).  After division we get a quotient of  (x² - xy + y²) with no remainder.  Therefore                        

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However, this method involves knowing the factor (x + y) beforehand (and the understanding of Factor Theorem). This article giao dịch with different methods of handling this factorization.

(Method 1) (Binomial theorem)

                  =

Move the last two terms to tướng the other side, we get:

                  =

(Method 2)

Move the last two terms to tướng the other side

=

(Method 3) (add a term and subtract the same term)

  =

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                  =

                  =

                   =

(Method 4)

Similar to (Method 3), you may start with subtracting a term and adding the same term: 

Can you continue with the factorization by grouping method?

(Method 5)      (change variable)

Consider  y = u V x   (1)

          =

          =,      by (1), u = x + y

          =

          =

M        Replacing hắn by (-y), you can get a new identity :

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(Exercise)    Prove

                  by following the methods above. Be careful to tướng note the sign of the identity.