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

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

(Method 1) (Binomial theorem)

=

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

=

(Method 2)

Move the last two terms to 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 – x (1)

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=, by (1), u = x + y

=

=

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

(Exercise) Prove

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