X3y3 + z3 can be factored by using a graphing calculator or through factoring as a sum of cubes. The final answer for factoring x3y3 + z3 is (xy + z) times (x2y2 - xyz + z2).
If a person is using a graphing calculator then he or she will need to make a change to the input by replacing z3 with z^3.
To factor by hand, a person will need to factor via the sum of cubes. In theory, a sum of two perfect cubes such as a3 + b3 can be factored into (a+b) times (a2 - ab + b2). The proof breaks down as follows: (a + b) times (a2 - ab + b2) = a3 - a2b + ab2 +ba2 - b2a + b3 = a3 + (a2b - ba2) + (ab2 - b2a) +b3 = a3 + 0 + 0 + b3 = a3 + b3.
In this instance, x3 is the cube of x1, y3 is the cube of y1 and z3 is the cube of z1. The factorization will therefore be written as (xy + z) times (x2y2 - xyz + z2). The next step is to see whether or not the multivariable polynomial (x2y2 - xyz + z2) can be factored. However, through the process of trial and error, the student will see that it is not factorable. Therefore, the simplified answer will remain (xy + z) times (x2y2 - xyz + z2).