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, which can be expanded using the formula \((a + b)^2 = a^2 + 2ab + b^2\).
2. Identify \(a = 9x\) and \(b = 7y\) in the expression \((9x + 7y)^2\).
3. Calculate \(a^2\), which is \((9x)^2 = 81x^2\).
4. Calculate \(2ab\), which is \(2 \times 9x \times 7y = 126xy\).
5. Calculate \(b^2\), which is \((7y)^2 = 49y^2\).

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. 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 and identify the degree of the polynomial is crucial for simplifying the final expression.