Factorization of x^{3} + y^{3} |
It
can be seen in most book that x^{3} + y^{3} can be factorized
by dividing the expression by (x
+ y). After division we get a
quotient of (x^{2} - xy
+ y^{2}) 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) = _{} =
_{} =
_{} =
_{} |
(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 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. |