In Exercises 9–20, find each product and write the result in standard form. (- 5 + 4i)(3 + i)

Verified step by step guidance

1. Identify the expression to be multiplied: $(-5+4i)(3+i)$.
2. Apply the distributive property (also known as the FOIL method for binomials): First, multiply the first terms: $-5\times 3$.
3. Next, multiply the outer terms: $-5\times i$.
4. Then, multiply the inner terms: $4i\times 3$.
5. Finally, multiply the last terms: $4i\times i$, and remember that $i^2=-1$.

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 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) to combine the real and imaginary parts. For example, when multiplying (-5 + 4i) and (3 + i), you multiply each part of the first complex number by each part of the second, carefully combining like terms and remembering that i^2 = -1.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. After performing operations on complex numbers, it is important to express the result in this form, ensuring that the real and imaginary components are clearly separated. This format is crucial for further mathematical operations and interpretations.