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.