Find each product. Write answers in standard form. (3-i)(3+1)(2-6i)

Verified step by step guidance

1. Step 1: First, we will multiply the first two terms together. To do this, we use the distributive property, which states that a(b + c) = ab + ac. So, (3 - i)(3 + 1) becomes 3*3 + 3*1 - i*3 - i*1.
2. Step 2: Simplify the expression obtained in step 1. Combine like terms and simplify the expression.
3. Step 3: Now, we will multiply the result from step 2 with the third term (2 - 6i). Again, we use the distributive property, which states that a(b + c) = ab + ac.
4. Step 4: Simplify the expression obtained in step 3. Combine like terms and simplify the expression.
5. Step 5: The result from step 4 is the product of the given terms. This is your final answer, and it should be in standard form, which is a + bi, where a and b are real numbers.

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, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding how to manipulate complex numbers is essential for solving problems involving them, 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) and combine like terms. When multiplying, remember that i^2 equals -1, which is crucial for simplifying the result. This process allows you to find the product of two or more complex numbers effectively.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. When solving problems involving complex numbers, it is important to express the final answer in this form for clarity and consistency. This involves ensuring that the real and imaginary parts are separated and properly simplified.