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

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 50

1장, 문제 50

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

검증된 단계별 안내

Recognize that the expression \((9 - 5x)^2\) represents a binomial squared. To expand this, use the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\).

Identify the terms in the binomial: \(a = 9\) and \(b = 5x\).

Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\): Substitute \(a = 9\) and \(b = 5x\) into the formula.

Calculate each term: \(a^2 = 9^2\), \(-2ab = -2(9)(5x)\), and \(b^2 = (5x)^2\).

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

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 form of the product.