In Exercises 15–58, find each product. (9−5x)²

Verified step by step guidance

1. Recognize that the expression \((9 - 5x)^2\) represents a binomial squared. To expand this, use the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\).
2. Identify the terms in the binomial: \(a = 9\) and \(b = 5x\).
3. Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\): Substitute \(a = 9\) and \(b = 5x\) into the formula.
4. Calculate each term: \(a^2 = 9^2\), \(-2ab = -2(9)(5x)\), and \(b^2 = (5x)^2\).
5. Combine the results from the previous step to write the expanded form of the expression.

Key Concepts

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, particularly those in the form of (a + b)^n. The expansion can be 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. In this case, (9 - 5x)^2 is a binomial expression that can be expanded using this theorem.

Squaring a Binomial

Squaring a binomial involves applying the formula (a - b)^2 = a^2 - 2ab + b^2. This formula allows us to find the square of a binomial expression by calculating the square of the first term, subtracting twice the product of the two terms, and adding the square of the second term. For (9 - 5x)^2, we will identify a as 9 and b as 5x to apply this formula.

Combining Like Terms

Combining like terms is a fundamental algebraic skill that involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. After expanding the expression (9 - 5x)^2, we will likely have multiple terms that can be simplified. This step is crucial for arriving at the final, simplified form of the product.