Find each product. See Example 5.(5r - 3t²)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.
3. Apply the formula: \((5r)^2 - 2(5r)(3t^2) + (3t^2)^2\).
4. Calculate each term separately: \((5r)^2 = 25r^2\), \(-2(5r)(3t^2) = -30rt^2\), and \((3t^2)^2 = 9t^4\).
5. Combine the terms to form the expanded expression: \(25r^2 - 30rt^2 + 9t^4\).

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)². The formula for the square of a binomial is (a + b)² = a² + 2ab + b². In the context of the given expression, (5r - 3t²)², it involves recognizing the two terms, 5r and -3t², and applying this formula to find the expanded form.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions using arithmetic operations and properties of equality. This includes distributing terms, combining like terms, and applying the distributive property. In the case of (5r - 3t²)², it requires careful multiplication of each term to ensure accuracy in the final expression.

Like Terms

Like terms are terms in an algebraic expression that have the same variable raised to the same power. They can be combined through addition or subtraction. When expanding (5r - 3t²)², identifying and combining like terms is essential to simplify the expression correctly, ensuring that the final result is in its simplest form.