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

Verified step by step guidance

1. Recognize that the expression is a product of two complex conjugates: \((3 + i)\) and \((3 - i)\).
2. Use the formula for the product of conjugates: \((a + b)(a - b) = a^2 - b^2\), where \(a = 3\) and \(b = i\).
3. Calculate \(a^2\): \(3^2 = 9\).
4. Calculate \(b^2\): \(i^2\). Recall that \(i^2 = -1\).
5. Substitute these values into the formula: \(9 - (-1)\), which simplifies to \(9 + 1\).

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 finding products.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL method) and applying the rule i² = -1 to simplify. For example, (3 + i)(3 - i) requires multiplying each term and combining like terms carefully.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. After multiplication, the result should be simplified and expressed in this form, separating the real and imaginary parts clearly.