Find each product. Write answers in standard form. (3+i)(3-i)

Verified step by step guidance

1. Recognize that the expression \((3+i)(3-i)\) is a product of conjugates.
2. Recall the formula for the product of conjugates: \((a+b)(a-b) = a^2 - b^2\).
3. Identify \(a = 3\) and \(b = i\) in the expression \((3+i)(3-i)\).
4. Apply the formula: \((3)^2 - (i)^2\).
5. Simplify the expression using \(i^2 = -1\) to write the result in standard form.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Multiplication of Complex Numbers

To multiply complex numbers, you apply the distributive property (also known as the FOIL method for binomials). For example, when multiplying (3+i)(3-i), you multiply each part of the first complex number by each part of the second, resulting in a combination of real and imaginary components that can be simplified.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' is the real part and 'b' is the imaginary part. After performing operations on complex numbers, it is important to simplify the result into this standard form for clarity and consistency in mathematical communication.