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

Verified step by step guidance

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

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 product of the expression.