Factor each polynomial completely. See Example 6. 6ar + 12br - 5as - 10bs

Verified step by step guidance

1. Identify the common factors in the polynomial terms. Notice that each term has a common factor of 1, and the terms can be grouped.
2. Group the terms in pairs: \((6ar + 12br)\) and \((-5as - 10bs)\).
3. Factor out the greatest common factor from each group: \(6r(a + 2b)\) and \(-5s(a + 2b)\).
4. Notice that \((a + 2b)\) is a common factor in both groups.
5. Factor out the common binomial factor \((a + 2b)\) from the expression: \((6r - 5s)(a + 2b)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. Common methods include factoring out the greatest common factor (GCF), using the difference of squares, and applying the quadratic formula for trinomials.

Greatest Common Factor (GCF)

The greatest common factor is the largest expression that divides all terms in a polynomial without leaving a remainder. Identifying the GCF is the first step in factoring, as it simplifies the polynomial and makes it easier to factor the remaining terms. For example, in the polynomial 6ar + 12br - 5as - 10bs, the GCF is 1, but grouping terms can also reveal common factors.

Grouping Method

The grouping method is a technique used to factor polynomials with four or more terms by grouping pairs of terms and factoring out common factors from each group. This method can help reveal hidden factors and simplify the polynomial. In the given polynomial, grouping terms strategically can lead to a complete factorization.