91Ó°ÊÓ

The first step in factoring any algebraic expression is identifying the greatest common factor (GCF). The GCF is the largest expression that can evenly divide all terms in the given polynomial. For the expression \(x^{12} - y^3 z^{12}\), there are no common variables or coefficients shared between the two terms, other than the number 1. Hence, the GCF here is 1. While identifying the GCF, ensure to look for common coefficients, variables, and their smallest exponents among the terms. This process simplifies the original expression and prepares it for further factorization.

After identifying the GCF, notice the structure of the expression. The given expression \(x^{12} - y^3 z^{12}\) can be seen as a 'difference of squares'. The difference of squares formula is given by \(a^2 - b^2 = (a - b)(a + b)\). Rewriting \(x^{12} - y^3 z^{12}\), we observe it resembles the pattern of a squared term minus another squared term. This allows us to apply the difference of squares method to factor it further. The difference of squares provides a useful way to break down complex polynomials into simpler binomials, making further factorization steps easier.

After recognizing and applying the difference of squares, we may end up with expressions that can still be factored. Complete factorization means continuing the process until we can no longer break down the expression. For instance, after the difference of squares is applied to \(x^{12}\), we might get additional simpler expressions such as \[ (x^{6} - y^{1.5} z^{6})(x^{6} + y^{1.5} z^{6}) \]. Each of these pieces should be examined to determine if they can be factored further. This process is iterated until we write the expression in its most simplified, fully factored form. By practicing complete factorization, students ensure that no further simplification is possible, achieving a thorough understanding of polynomial factorization.