Find each product. Write answers in standard form. (2+i)²

Verified step by step guidance

1. Recall the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).
2. Identify \(a = 2\) and \(b = i\) in the expression \((2 + i)^2\).
3. Apply the formula: \((2 + i)^2 = 2^2 + 2 \cdot 2 \cdot i + i^2\).
4. Calculate each term separately: \(2^2 = 4\), \(2 \cdot 2 \cdot i = 4i\), and recall that \(i^2 = -1\).
5. Combine the terms to write the expression in standard form: \(4 + 4i + (-1)\), then simplify to \(3 + 4i\).

Key Concepts

Complex Numbers

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. Understanding how to work with complex numbers is essential for operations like addition, multiplication, and exponentiation.

Binomial Expansion

Binomial expansion involves expanding expressions raised to a power, such as (a + b)² = a² + 2ab + b². This formula helps simplify powers of binomials, including those with complex terms, by systematically multiplying and combining like terms.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part. Writing answers in standard form means expressing the result clearly with real and imaginary parts separated, which is important for clarity and further calculations.