Find each product. [(3q+5)-p][(3q+5)+p]

검증된 단계별 안내

Recognize that the expression \([(3q+5)-p][(3q+5)+p]\) is in the form of a product of conjugates, which follows the pattern \((a - b)(a + b) = a^2 - b^2\).

Identify \(a = (3q + 5)\) and \(b = p\) from the given expression.

Apply the difference of squares formula: \((3q + 5)^2 - p^2\).

Expand \((3q + 5)^2\) by using the formula \((x + y)^2 = x^2 + 2xy + y^2\), where \(x = 3q\) and \(y = 5\).

Write the final expression as \( (3q)^2 + 2 \cdot 3q \cdot 5 + 5^2 - p^2 \) and simplify each term without calculating the final numeric values.

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주요 개념

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

Difference of Squares

The difference of squares is a special product formula expressed as (a - b)(a + b) = a² - b². It allows quick multiplication of two binomials that are conjugates, simplifying the expression by subtracting the square of the second term from the square of the first.

Binomial Expressions

A binomial is an algebraic expression containing two terms connected by addition or subtraction, such as (3q + 5) or p. Understanding how to manipulate and multiply binomials is essential for expanding and simplifying algebraic expressions.

Polynomial Multiplication

Polynomial multiplication involves applying the distributive property to multiply each term in one polynomial by every term in the other. This process is fundamental for expanding products of binomials and simplifying the resulting expressions.