Find each product. See Examples 5 and 6. [(3a+b)-1]^2

Verified step by step guidance

1. Identify the expression to be squared: \((3a + b) - 1\).
2. Recognize that this is a binomial squared, which can be expanded using the formula \((x - y)^2 = x^2 - 2xy + y^2\).
3. In this case, let \(x = 3a + b\) and \(y = 1\).
4. Apply the formula: \(((3a + b) - 1)^2 = (3a + b)^2 - 2(3a + b)(1) + 1^2\).
5. Expand each term: \((3a + b)^2\), \(-2(3a + b)\), and \(1^2\), then combine like terms.

Key Concepts

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power, particularly those in the form of (a + b)^n. The expansion is based on the Binomial Theorem, which states that (x + y)^n can be expressed as a sum of terms involving coefficients from Pascal's Triangle. Understanding this concept is crucial for simplifying expressions like [(3a + b) - 1]^2.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions using various operations such as addition, subtraction, multiplication, and division. This skill is essential for solving equations and simplifying complex expressions. In the context of the given question, it helps in correctly applying the square to the binomial and simplifying the result.

Exponent Rules

Exponent rules are a set of mathematical guidelines that dictate how to handle expressions involving powers. Key rules include the product of powers, power of a power, and power of a product. These rules are vital when expanding expressions like [(3a + b) - 1]^2, as they help in correctly applying the exponent to each term in the binomial.