Find each product. See Examples 5 and 6. (5r-3t^2)^2

Verified step by step guidance

1. Recognize that the expression \((5r - 3t^2)^2\) is a binomial squared, which can be expanded using the formula \((a - b)^2 = a^2 - 2ab + b^2\).
2. Identify \(a = 5r\) and \(b = 3t^2\) in the expression \((5r - 3t^2)^2\).
3. Calculate \(a^2\) which is \((5r)^2 = 25r^2\).
4. Calculate \(2ab\) which is \(2 \times 5r \times 3t^2 = 30rt^2\).
5. Calculate \(b^2\) which is \((3t^2)^2 = 9t^4\).

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 achieved using the Binomial Theorem, which states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This concept is essential for solving the given problem as it allows us to expand the squared binomial expression.

Squaring a Binomial

Squaring a binomial involves multiplying the binomial by itself. For a binomial of the form (a + b), the square is calculated as (a + b)^2 = a^2 + 2ab + b^2. In the context of the question, we need to apply this principle to the expression (5r - 3t^2)^2, which will require careful attention to the signs and coefficients during multiplication.

Combining Like Terms

Combining like terms is a fundamental algebraic process that involves simplifying expressions by adding or subtracting terms that have the same variable components. After expanding the binomial, it is crucial to identify and combine any like terms to arrive at the final simplified expression. This step ensures that the result is presented in its simplest form, making it easier to interpret and use.