Factorization is a fundamental mathematical process where we break down complex expressions into simpler parts called factors. Think of it like taking apart a machine to see its components. For numbers, we find what multiplies together to give the original number. For polynomials, we find simpler expressions that multiply to give the original polynomial. This skill is essential for solving equations and simplifying complex mathematical expressions.
The first and most important step in factorization is finding the Greatest Common Factor, or GCF. Look at each term in your expression and identify what factors they all share. This includes both numerical coefficients and variable parts. In our example, six x cubed plus nine x squared plus twelve x, we can see that all terms contain the factor three x. We factor this out to get three x times the quantity two x squared plus three x plus four. Always start with the GCF as it simplifies the remaining expression.
The second major technique is factoring trinomials, especially quadratics. For a trinomial like x squared plus five x plus six, we need to find two numbers that multiply to give the constant term and add to give the middle coefficient. In this case, we need numbers that multiply to six and add to five. The numbers two and three work perfectly: two times three equals six, and two plus three equals five. Therefore, our factored form is x plus two times x plus three.
Special product formulas are powerful shortcuts for factorization. The difference of squares pattern, a squared minus b squared, always factors as a minus b times a plus b. For example, x squared minus nine becomes x minus three times x plus three. Perfect square trinomials follow the pattern a squared plus or minus two a b plus b squared, which factors as a plus or minus b squared. Recognizing these patterns instantly speeds up your factorization process significantly.
Let's complete a full example to summarize our factorization process. Starting with twelve x cubed minus forty-eight x, we first find the GCF, which is twelve x. This gives us twelve x times the quantity x squared minus four. We recognize x squared minus four as a difference of squares pattern, so it factors as x minus two times x plus two. Our final answer is twelve x times x minus two times x plus two. Always check your work by multiplying the factors back together to verify you get the original expression. With practice, these steps become automatic and factorization becomes much easier.