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

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 74

1장, 문제 74

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

검증된 단계별 안내

Recognize that the expression \((9x + 7y)^2\) is a binomial squared. This means you will use the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).

Identify the terms in the binomial: \(a = 9x\) and \(b = 7y\).

Apply the formula \((a + b)^2 = a^2 + 2ab + b^2\) to the given expression. Substitute \(a = 9x\) and \(b = 7y\) into the formula.

Calculate each term separately: \(a^2 = (9x)^2\), \(2ab = 2(9x)(7y)\), and \(b^2 = (7y)^2\).

Combine the results from the previous step to write the expanded form of the expression.

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주요 개념

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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.