Factor out the greatest common factor from each polynomial. See Example 1. (5r-6)(r+3)-(2r-1)(r+3)

Verified step by step guidance

1. Identify the common factor in both terms of the expression: (5r - 6)(r + 3) - (2r - 1)(r + 3). Notice that (r + 3) appears in both terms.
2. Factor out the common binomial factor (r + 3) from the entire expression. This means rewriting the expression as (r + 3) multiplied by the difference of the remaining factors.
3. Write the expression as: \( (r + 3) \left[ (5r - 6) - (2r - 1) \right] \).
4. Simplify the expression inside the brackets by distributing the negative sign to the second group: \( (5r - 6) - (2r - 1) = 5r - 6 - 2r + 1 \).
5. Combine like terms inside the brackets: \( (5r - 2r) + (-6 + 1) = 3r - 5 \). So the factored expression is \( (r + 3)(3r - 5) \).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest expression that divides two or more terms or polynomials without leaving a remainder. Factoring out the GCF simplifies expressions by extracting this common part, making further operations easier.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It allows you to factor expressions by reversing this process, identifying common factors across terms and rewriting the expression as a product of factors.

Polynomial Multiplication and Simplification

Multiplying polynomials involves applying the distributive property to each term and combining like terms. Simplifying the resulting expression helps identify common factors and prepares the polynomial for factoring.