Factorization is the process of breaking down algebraic expressions into their multiplicative components. For example, 6x plus 9 can be factored as 3 times the quantity 2x plus 3. We can visualize this using rectangles where the area represents our expression, and factorization finds the dimensions. This process is the inverse of multiplication, helping us simplify and solve algebraic problems.
The common factor method systematically finds the greatest common factor in polynomial expressions. First, identify the numerical GCF of all coefficients. Then find the variable GCF by taking the lowest power of each variable. Finally, factor out this GCF. For example, in 12x squared y plus 18xy squared plus 6xy, the GCF is 6xy, giving us 6xy times the quantity 2x plus 3y plus 1.
Factoring by grouping handles four-term polynomials without obvious common factors. We strategically pair terms into groups, factor each group separately, then identify the common binomial factor. For example, ax plus ay plus bx plus by becomes a times x plus y, plus b times x plus y, which factors as a plus b times x plus y. This method requires recognizing the common binomial pattern.
Quadratic expressions follow three main factorization patterns. The difference of squares pattern, a squared minus b squared, factors as a plus b times a minus b. Perfect square trinomials factor as perfect squares. General trinomials require systematic approaches like the AC method. For example, x squared minus 9 factors as x plus 3 times x minus 3, demonstrating the difference of squares pattern.
Advanced factorization combines multiple techniques systematically. Sum and difference of cubes follow specific formulas. Complex expressions require step-by-step factoring, like 2x to the fourth minus 32, which factors completely as 2 times x minus 2 times x plus 2 times x squared plus 4. Factorization enables solving equations by setting each factor to zero. Always follow a systematic strategy: check for common factors first, then special patterns, then grouping methods, and verify complete factorization.