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

Verified step by step guidance

1. Recognize that the expression \((x+5)^2\) represents a binomial squared. This means you will apply the formula for squaring a binomial: \((a+b)^2 = a^2 + 2ab + b^2\).
2. Identify the terms in the binomial: \(a = x\) and \(b = 5\). Substitute these values into the formula.
3. Expand the squared binomial using the formula: \((x+5)^2 = x^2 + 2(x)(5) + 5^2\).
4. Simplify each term: \(x^2\) remains as is, \(2(x)(5)\) simplifies to \(10x\), and \(5^2\) simplifies to \(25\).
5. Combine all the simplified terms to express the expanded form of the binomial: \(x^2 + 10x + 25\).

Key Concepts

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.

Perfect Square Trinomial

A perfect square trinomial is an expression that can be written in the form (a + b)^2 = a^2 + 2ab + b^2. In the case of (x + 5)^2, it represents the square of the binomial x + 5. Recognizing this form allows for quick expansion without needing to multiply the binomial by itself directly.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions using established rules and properties. This includes operations such as distribution, combining like terms, and applying the laws of exponents. Mastery of these techniques is essential for effectively expanding and simplifying expressions like (x + 5)^2.