In Exercises 35–44, factor the greatest common binomial factor from each polynomial.3x(x+y) − (x+y)

Verified step by step guidance

1. Identify the common binomial factor in the expression. In this case, the common binomial factor is \((x+y)\).
2. Factor out the common binomial factor \((x+y)\) from each term in the polynomial.
3. Rewrite the expression as \((x+y)(3x - 1)\).
4. Verify the factorization by distributing \((x+y)\) back into \((3x - 1)\) to ensure it equals the original expression.
5. Conclude that the expression is factored correctly as \((x+y)(3x - 1)\).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest expression that divides two or more terms without leaving a remainder. In polynomial expressions, identifying the GCF involves finding the highest degree of common variables and coefficients among the terms. This concept is crucial for simplifying expressions and factoring polynomials effectively.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which can include numbers, variables, or binomials. This process is essential for solving polynomial equations and simplifying expressions. Understanding how to factor correctly allows for easier manipulation and analysis of algebraic expressions.

Binomial Expressions

A binomial expression is a polynomial that consists of exactly two terms, typically separated by a plus or minus sign. In the context of the given question, recognizing the binomial factor (x+y) is key to factoring the entire polynomial. Understanding binomials is fundamental in algebra as they frequently appear in various algebraic operations and equations.