Problem 65 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (3 i)(2-5 i) $$

Step-by-Step Solution

Answer

The product is 15 + 6i.

Step 1: Apply Distributive Property

To find the product (3i)(2-5i), we distribute 3i over the other terms: 3i * 2 and 3i * -5i.

Step 2: Simplify Each Term

Calculate the products: - For 3i * 2, the result is 6i. - For 3i * -5i, the result is -15i^2, where i^2 = -1.

Step 3: Recognize i^2 and Simplify

Since i^2 = -1, replace -15i^2 with -15(-1), which results in +15.

Step 4: Combine Real and Imaginary Parts

Now, add together the real and imaginary parts: 15 + 6i.

Step 5: Express in Standard Form

The standard form of a complex number is a + bi. Therefore, the expression 15 + 6i is already in standard form, where a = 15 and b = 6.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra, pushing us to rethink multiplication as a form of addition. When applied to complex numbers, it allows us to multiply a single term by each term within parentheses. This is especially useful in expressions like \((3i)(2-5i)\), where applying the distributive property involves spreading the factor outside the parentheses across each term inside.

Here's how it works:

• Multiply \(3i\) by \(2\), giving us \(6i\).
• Then, multiply \(3i\) by \(-5i\), resulting in \(-15i^2\).

By understanding the distributive property, you can simplify complex multiplications one step at a time, making big problems feel smaller and more manageable.

Imaginary Unit

The imaginary unit, represented as \(i\), is a special element in mathematics. It's defined by the property \(i^2 = -1\). Although it might sound a bit strange at first, it's an incredibly useful tool for expanding our understanding of numbers and solving equations that don't have real solutions.

Here's why the imaginary unit matters:

• In mathematics, especially algebra, real numbers are insufficient to explain solutions to all polynomial equations.
• The imaginary unit helps us define new numbers, known as complex numbers, which incorporate both real and imaginary components.
• For example, when multiplying by \(-15i^2\), knowing \(i^2 = -1\) allows us to convert it to real number form: \(-15(-1) = 15\).

Through the lens of \(i\), we can transition between real and imaginary realms smoothly, bridging gaps that real numbers alone can't fill.

Standard Form of a Complex Number

Every complex number can be neatly written in a style known as the standard form: \(a + bi\). Here, \(a\) is the real part, while \(b\) represents the coefficient of the imaginary part \(i\).

Standard form makes complex numbers easy to understand and work with because:

• It clearly shows the relationship between the real and imaginary parts.
• In problems, like converting \(15 + 6i\), this form helps identify the components \(a = 15\) and \(b = 6\).
• It serves as a common language for mathematicians, facilitating communication and problem-solving.

Recognizing and using the standard form simplifies expressions and equations, making calculations with complex numbers straightforward and intuitive.