In Exercises 67–82, find each product. (9x+7y)²

Verified step by step guidance

1. Recognize that the expression \((9x + 7y)^2\) is a binomial squared. 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 in the binomial: \(a = 9x\) and \(b = 7y\).
3. Apply the formula \((a + b)^2 = a^2 + 2ab + b^2\) to the given expression. Substitute \(a = 9x\) and \(b = 7y\) into the formula.
4. Calculate each term separately: \(a^2 = (9x)^2\), \(2ab = 2(9x)(7y)\), and \(b^2 = (7y)^2\).
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. This theorem provides a systematic way to calculate the coefficients of the expanded terms.

Squaring a Binomial

Squaring a binomial involves multiplying the binomial by itself. For a binomial (a + b), the square is calculated as (a + b)(a + b), which results in a^2 + 2ab + b^2. This formula is essential for simplifying expressions like (9x + 7y)^2, as it allows for the direct computation of the resulting polynomial.

Polynomial Terms

Polynomial terms are expressions that consist of variables raised to non-negative integer powers, multiplied by coefficients. In the context of the expression (9x + 7y)^2, the resulting polynomial will contain terms such as x^2, xy, and y^2, each representing different degrees of the variables. Understanding how to combine like terms is crucial for simplifying the final expression.