Problem 57 - A Graphical Approach to College Algebra

Question

Multiply as indicated. Write each product in standard form. $$(2+4 i)(-1+3 i)$$

Step-by-Step Solution

Answer

The product is \(-14 + 2i\).

Step 1: Apply the Distributive Property

To multiply the complex numbers \((2+4i)(-1+3i)\), begin by using the distributive property. This involves multiplying each term in the first complex number by each term in the second complex number.So multiply:\[ (2)(-1), (2)(3i), (4i)(-1), (4i)(3i) \]

Step 2: Multiply the Real and Imaginary Components

Calculate the four separate products from the previous step: 1. \(2 \times -1 = -2 \)2. \(2 \times 3i = 6i \)3. \(4i \times -1 = -4i \)4. \(4i \times 3i = 12i^2 \)

Step 3: Simplify the Complex Number

Next, combine the calculated terms together: \( -2 + 6i - 4i + 12i^2 \)Note that \(i^2 = -1\), thus \(12i^2 = 12(-1) = -12\). Substitute this value to get:\( -2 + 6i - 4i - 12 \)

Step 4: Combine Like Terms

Finally, simplify by combining like terms. Start by combining the real numbers:\(-2 - 12 = -14\)Then combine the imaginary numbers:\(6i - 4i = 2i\)Thus, the product in standard form is:\(-14 + 2i\)

Key Concepts

Distributive Property

When multiplying complex numbers such as \((2+4i)(-1+3i)\),we use the Distributive Property.This property tells us to multiply each term in the first complex number by each term in the second. Think of it like distributing each part of one number across every part of the other.
To illustrate:

• First, multiply \(2\) by each term in \((-1 + 3i)\).You'll get \(-2\) and \(6i\).
• Next, multiply \(4i\) by each term in \((-1 + 3i)\).This yields \(-4i\) and \(12i^2\).

By systematically distributing each part, we ensure every combination is covered, allowing us to correctly sum all the products.
This approach is fundamental in finding the product of two complex numbers.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers.In this exercise, after distributing and simplifying, we aim to write our solution in this form.
After multiplication and simplification,we end up with real and imaginary components, such as:

• \(-14\) (the real part)
• \(2i\) (the imaginary part)

Thus, the final step is to combine these components into the standard form:\(-14 + 2i\).Positioning real parts before imaginary parts maintains consistency and clarity in mathematical communication.

Imaginary Units

Imaginary units are key in understanding complex numbers.The unit \(i\) represents the square root of \(-1\).When working with \(i\), it's important to remember that \(i^2 = -1\).
This property was crucial when simplifying \(4i \times 3i = 12i^2\).Using \(i^2 = -1\), we transformed \(12i^2\) into \(-12\).
Understanding how imaginary units combine and simplify is vital when working with complex multiplication.This comprehension helps keep calculations correct and ensures results are expressed in the simplest terms.