Find each product. See Example 5. (5r - 3t²)²

Verified step by step guidance

1. Recognize that the expression is a square of a binomial: \((5r - 3t^{2})^{2}\). This means you will use the formula for the square of a binomial: \((a - b)^{2} = a^{2} - 2ab + b^{2}\).
2. Identify the terms \(a\) and \(b\) in the binomial: here, \(a = 5r\) and \(b = 3t^{2}\).
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 Expansion

Binomial expansion involves expanding expressions raised to a power, such as (a + b)². It follows the formula (a + b)² = a² + 2ab + b², which helps in simplifying and finding the product of binomials.

Polynomial Multiplication

Polynomial multiplication requires multiplying each term in one polynomial by every term in the other. This process ensures all products are accounted for and combined correctly to simplify the expression.

Exponents and Like Terms

Understanding exponents is crucial when squaring terms, as powers multiply (e.g., (t²)² = t⁴). Combining like terms after expansion simplifies the expression by adding coefficients of terms with the same variables and exponents.