Problem 73 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (-3-2 i)(5+6 i) $$

Step-by-Step Solution

Answer

The product is \(-27 - 28i\), a complex number in standard form.

Step 1: Apply the Distributive Property

To find the product of \((-3-2i)(5+6i)\), apply the distributive property (also known as the FOIL method for binomials):\((-3)(5) + (-3)(6i) + (-2i)(5) + (-2i)(6i)\)

Step 2: Compute Each Term

Calculate each part of the expression:- \((-3)(5) = -15\)- \((-3)(6i) = -18i\)- \((-2i)(5) = -10i\)- \((-2i)(6i) = 12i^2\)Recall that \(i^2 = -1\), so \(12i^2 = 12(-1) = -12\).

Step 3: Add the Real and Imaginary Parts

Combine the real parts and the imaginary parts:- Real parts: \(-15 - 12 = -27\)- Imaginary parts: \(-18i - 10i = -28i\).

Step 4: Write in Standard Form

Combine the results from Step 3 to express the product in standard form: \(-27 - 28i\), where \(a = -27\) and \(b = -28\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term by each term in a parenthesis. It is especially useful when dealing with expressions like \((a + b)(c + d)\), where you distribute or "spread out" one binomial term over the other. In this context, it is referred to as the FOIL method when applied to binomials.
Imagine unpacking a boxed set of values. Each term from one set has to pair with every term from the other set. This is akin to distribution, ensuring that each component is accounted for. For example, given \((-3-2i)(5+6i)\), you break it down as follows:

• Multiply \(-3\) with both \(5\) and \(6i\)
• Multiply \(-2i\) with both \(5\) and \(6i\)

This process is methodical and ensures a comprehensive product of the terms involved.
Ultimately, applying this property simplifies the expression and prepares it for final computation, setting the stage for more sophisticated techniques, such as finding the standard form of complex numbers.

FOIL Method

The FOIL method is a mnemonic for remembering the process of multiplying two binomials. The letters stand for First, Outer, Inner, and Last, describing which terms in each binomial should be multiplied together.
In the expression \((-3 - 2i)(5 + 6i)\), the FOIL method involves:

• **First**: Multiply the first terms: \(-3 \times 5 = -15\)
• **Outer**: Multiply the outer terms: \(-3 \times 6i = -18i\)
• **Inner**: Multiply the inner terms: \(-2i \times 5 = -10i\)
• **Last**: Multiply the last terms: \(-2i \times 6i = 12i^2\)

After executing these steps, don't forget an important characteristic of imaginary numbers: \(i^2\) equals \(-1\). This identity helps in simplifying complex expressions, converting \(12i^2\) to \(-12\).
The FOIL method is a systematic approach ensuring you consider all interactions between the components of the binomials. It is especially useful in handling complex numbers where careful attention to both real and imaginary parts is crucial.

Standard Form of a Complex Number

The standard form of a complex number is essential for expressing results clearly and concisely. This format is denoted as \(a + bi\), where \(a\) represents the real part and \(bi\) the imaginary part.
Complex numbers arise from using \(i\), the imaginary unit, defined such that \(i^2 = -1\). In the given exercise, after applying the distributive property and the FOIL method, the product is initially broken into real and imaginary segments:

• Real parts: \(-15\) and \(-12\), resulting in \(-27\)
• Imaginary parts: \(-18i\) and \(-10i\), leading to \(-28i\)

Combining these parts provides the standard form: \(-27 - 28i\).
Using this form ensures clarity and uniformity across mathematical computations, allowing easy identification of the real and imaginary elements. Understanding complex numbers in standard form also simplifies operations like addition, subtraction, multiplication, and division, making it an invaluable tool in advanced areas of mathematics.