Factor each polynomial by grouping. See Example 2. 10ab-6b+35a-21

Verified step by step guidance

1. Group the terms in pairs to prepare for factoring by grouping: \( (10ab - 6b) + (35a - 21) \).
2. Factor out the greatest common factor (GCF) from each group separately: from the first group, factor out \$2b\( to get \)2b(5a - 3)\(; from the second group, factor out \(7\) to get \)7(5a - 3)$.
3. Notice that both groups contain the common binomial factor \( (5a - 3) \).
4. Factor out the common binomial factor \( (5a - 3) \) from the entire expression: \( (5a - 3)(2b + 7) \).
5. Write the final factored form as the product of the two factors: \( (5a - 3)(2b + 7) \).

Key Concepts

Polynomial Terms and Like Terms

Polynomials are algebraic expressions made up of terms consisting of variables and coefficients. Understanding how to identify and group like terms, which share the same variables raised to the same powers, is essential for factoring by grouping.

Factoring by Grouping

Factoring by grouping involves rearranging and grouping terms in a polynomial to find common factors within each group. This method simplifies the polynomial into a product of binomials or other factors by extracting the greatest common factor from each group.

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest expression that divides two or more terms without leaving a remainder. Identifying the GCF in each group is crucial for factoring by grouping, as it allows you to factor out common elements and simplify the polynomial.