Find each product. (5r-3t2)2

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 49

1장, 문제 49

Find each product. (5r-3t²)²

검증된 단계별 안내

Recognize that the expression \((5r - 3t^2)^2\) is a binomial squared, which can be expanded using the formula \((a - b)^2 = a^2 - 2ab + b^2\).

Identify \(a = 5r\) and \(b = 3t^2\) in the expression \((5r - 3t^2)^2\).

Calculate the square of the first term: \(a^2 = (5r)^2 = 25r^2\).

Calculate twice the product of the two terms: \(-2ab = -2 \times 5r \times 3t^2 = -30rt^2\).

Calculate the square of the second term: \(b^2 = (3t^2)^2 = 9t^4\), then combine all parts to write the expanded expression as \(25r^2 - 30rt^2 + 9t^4\).

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주요 개념

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

Binomial Squaring Formula

The binomial squaring formula states that (a - b)^2 = a^2 - 2ab + b^2. This formula helps expand the square of a binomial expression by squaring each term and subtracting twice their product, simplifying the process of finding the product.

Exponent Rules

Exponent rules govern how to handle powers in algebraic expressions, such as (x^m)^n = x^(m*n) and the product rule x^m * x^n = x^(m+n). Understanding these rules is essential for correctly simplifying terms like t^2 when squaring the binomial.

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in the other. For binomials, this means applying the distributive property twice or using special formulas like the binomial square to find the product efficiently.