Find each product. Write answers in standard form. (1+3i)(2-5i)

Verified step by step guidance

1. Recall that to multiply two complex numbers, we use the distributive property (also known as FOIL for binomials): multiply each term in the first complex number by each term in the second complex number.
2. Write the expression explicitly: \((1 + 3i)(2 - 5i) = 1 \cdot 2 + 1 \cdot (-5i) + 3i \cdot 2 + 3i \cdot (-5i)\).
3. Calculate each product separately: \(1 \cdot 2 = 2\), \(1 \cdot (-5i) = -5i\), \(3i \cdot 2 = 6i\), and \(3i \cdot (-5i) = -15i^2\).
4. Combine the like terms: the real parts \(2\) and \(-15i^2\), and the imaginary parts \(-5i\) and \$6i\(. Remember that \)i^2 = -1\(, so replace \)i^2\( with \)-1$ in the expression.
5. Simplify the expression by calculating \(-15i^2\) as \(-15(-1)\), then add the real parts and the imaginary parts separately to write the final product in standard form \(a + bi\).

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. They extend the real number system and are used to represent quantities involving the square root of negative numbers.

Multiplication of Complex Numbers

To multiply complex numbers, use the distributive property (FOIL method) to expand the product, then combine like terms. Remember to apply i² = -1 to simplify terms involving i² into real numbers.

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. After multiplication, simplify the expression so it clearly separates the real and imaginary components in this form.