Find each product. (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 \((5r - 3t^2)^2\).
3. Calculate the square of the first term: \(a^2 = (5r)^2 = 25r^2\).
4. Calculate twice the product of the two terms: \(-2ab = -2 \times 5r \times 3t^2 = -30rt^2\).
5. Calculate the square of the second term: \(b^2 = (3t^2)^2 = 9t^4\), then combine all parts to write the expanded expression as \(25r^2 - 30rt^2 + 9t^4\).

Key Concepts

Binomial Squaring Formula

The binomial squaring formula states that (a - b)^2 = a^2 - 2ab + b^2. This formula helps expand the square of a binomial expression by squaring each term and subtracting twice their product, simplifying the process of finding the product.

Exponent Rules

Exponent rules govern how to handle powers in algebraic expressions, such as (x^m)^n = x^(m*n) and the product rule x^m * x^n = x^(m+n). Understanding these rules is essential for correctly simplifying terms like t^2 when squaring the binomial.

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in the other. For binomials, this means applying the distributive property twice or using special formulas like the binomial square to find the product efficiently.