Perform the indicated operations. (3p+5)²

Verified step by step guidance

1. Recognize that the expression \( (3p + 5)^2 \) is a binomial squared, which means it can be expanded using the formula for the square of a binomial: \( (a + b)^2 = a^2 + 2ab + b^2 \).
2. Identify the terms in the binomial: here, \( a = 3p \) and \( b = 5 \).
3. Calculate the square of the first term: \( (3p)^2 = 9p^2 \).
4. Calculate twice the product of the two terms: \( 2 \times (3p) \times 5 = 30p \).
5. Calculate the square of the second term: \( 5^2 = 25 \). Then, combine all parts to write the expanded form: \( 9p^2 + 30p + 25 \).

Key Concepts

Binomial Expansion

Binomial expansion involves expressing the power of a binomial, such as (a + b)^n, as a sum of terms using the Binomial Theorem or by applying algebraic identities. For the square of a binomial, (a + b)^2, the expansion is a^2 + 2ab + b^2.

Algebraic Identities

Algebraic identities are formulas that simplify the process of expanding expressions. The identity (a + b)^2 = a^2 + 2ab + b^2 is essential for quickly expanding squared binomials without multiplying each term individually.

Polynomial Operations

Polynomial operations include addition, multiplication, and simplification of polynomial expressions. Understanding how to multiply terms and combine like terms is crucial when expanding and simplifying expressions like (3p + 5)^2.