Factoring by grouping is a technique used to factor polynomials with four or more terms. In the given example, 15 x3 – 5x2 + 6x – 2, the terms are grouped into pairs: (15 x3 – 5x2) and (6x – 2). The greatest common factor (GCF) is then extracted from each pair. The GCF of the first pair is 5 x2, resulting in 5x2(3x – 1). The GCF of the second pair is 2, resulting in 2(3x – 1). Since both resulting expressions share a common binomial factor, (3x – 1), it can be further factored out, yielding the final factored form: (3x – 1)(5*x2 + 2).
This method simplifies complex polynomial expressions into more manageable forms. This simplification is crucial in various mathematical operations, including solving equations, finding roots, and simplifying rational expressions. Factoring reveals the underlying structure of a polynomial, providing insights into its behavior and properties. Historically, factoring techniques have been essential tools in algebra, contributing to advancements in numerous fields, including physics, engineering, and computer science.
