Problem 52 - Intermediate Algebra

Question

Multiply. Write all answers in the form \(a+b i.\) $$ -5(3+2 i) $$

Step-by-Step Solution

Answer

The product is \\(-15 - 10i\\).

Step 1: Distribute the Real Part

To find the product, distribute \(-5\) across the real part of \(3+2i\). Multiply \(-5\) by \(3\): \(-5 \times 3 = -15\).

Step 2: Distribute the Imaginary Part

Next, distribute \(-5\) across the imaginary part of \(3+2i\). Multiply \(-5\) by \(2i\):\(-5 \times 2i = -10i\).

Step 3: Combine the Results

Combine the two parts we obtained from the distribution in the previous steps. This gives us the final expression:\(-15 - 10i\).

Key Concepts

Multiplication of Complex Numbers

Complex numbers are a fundamental concept in advanced mathematics that combine real and imaginary numbers to expand the number system we learn early in school. A complex number typically looks like this: \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. You might wonder how multiplication fits into this framework. In multiplication of complex numbers, each component of the number, both the real and the imaginary, is multiplied separately and then combined.
To multiply complex numbers, each term in the first expression is multiplied by each term in the second. This operation is similar to multiplying polynomials. When dealing with a single complex number like \(-5(3 + 2i)\), you use the same idea, multiplying the real number by each part of the complex number, as shown in our exercise example.

• First, multiply the real part \(-5\) with the real part of the complex number, \(3\).
• Then, multiply the real part \(-5\) with the imaginary part, \(2i\).
• Add the results together.

Distributive Property

The distributive property is a crucial principle in algebra, and it is essential when multiplying complex numbers. It allows us to break down a multiplicative process into more straightforward parts, making complex multiplication manageable.
When we distribute in algebra, we multiply a single term across each term within parentheses in another polynomial or expression. Here’s how it works with complex numbers: when you encounter an expression like \(-5(3 + 2i)\), you distribute \(-5\) to both \(3\) and \(2i\).
To apply the distributive property:

• Multiply \(-5\) by each part inside the parentheses individually.
• Use the results from these operations to form the simplified expression.

The distributive property simplifies calculations and is a key tactic in handling potentially confusing imaginary components.

Imaginary Numbers

Imaginary numbers bring an interesting twist to the world of mathematics. They are usually denoted as \(i\), where \(i\) is the square root of \(-1\). This concept at first may seem counterintuitive since we are not accustomed to taking square roots of negative numbers.
In dealing with complex numbers, understanding \(i\) is imperative. Essentially, any real number multiplied by \(i\) will yield an imaginary number. For instance, in \(3 + 2i\), the \(2i\) represents the imaginary part.
In multiplication, especially like in our step-by-step solution, where we multiply \(-5\) by \(2i\), we need to remember:

• Multiplying a real number by \(i\) results in an imaginary number.
• Keep track of the sign since \(i\) affects the terms differently than regular numbers.

Our purpose is to navigate operations involving \(i\) carefully, always understanding that \(i^2 = -1\), which will be vital when more complex multiplications arise.