Problem 25 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (-1+2 i)(-2+3 i) $$

Step-by-Step Solution

Answer

The simplified complex number is \(-4 - 7i\).

Step 1: Apply the Distributive Property

We perform the multiplication by applying the distributive property, which is also referred to as the FOIL method for binomials. Expand the expression \((-1 + 2i)(-2 + 3i)\) to: \[(-1)(-2) + (-1)(3i) + (2i)(-2) + (2i)(3i)\]

Step 2: Multiply Each Term

Calculate each part of the expansion: 1) \((-1)(-2) = 2\)2) \((-1)(3i) = -3i\)3) \((2i)(-2) = -4i\)4) \((2i)(3i) = 6i^2\)

Step 3: Simplify Using \(i^2 = -1\)

Since \(i^2 = -1\), we simplify the term \(6i^2\): \(6i^2 = 6(-1) = -6\).So the expression becomes: \[2 - 3i - 4i - 6\]

Step 4: Combine Like Terms

Combine the real parts and the imaginary parts separately: Real part: \(2 - 6 = -4\)Imaginary part: \(-3i - 4i = -7i\)Thus, the simplified form of the expression is: \(-4 - 7i\).

Key Concepts

Distributive Property

The distributive property is a basic principle of algebra that allows us to multiply a single term by each term inside a set of parentheses. It states that for any numbers or expressions \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. This rule helps in expanding expressions and is crucial in simplifying complex numbers.

When dealing with complex numbers like \((-1 + 2i)(-2 + 3i)\), the distributive property allows us to expand the expression by multiplying each term in the first binomial with each term in the second:

• The first term of the first binomial (\(-1\)) multiplies both terms in the second binomial.
• Similarly, the second term (\(2i\)) does the same.

This step-by-step expansion makes complex multiplication easier and sets a clear path for further calculation.

FOIL Method

The FOIL method is a specific application of the distributive property for binomial expressions. FOIL stands for First, Outer, Inner, and Last, indicating the order of terms to be multiplied in each binomial pair.

Using the FOIL method for the expression \((-1 + 2i)(-2 + 3i)\):

• First: Multiply the first terms from each binomial, i.e., \((-1)(-2)\).
• Outer: Multiply the outer terms \((-1)(3i)\).
• Inner: Multiply the inner terms \((2i)(-2)\).
• Last: Multiply the last terms from each binomial \((2i)(3i)\).

The FOIL method streamlines the process and ensures that all interactions between terms are accounted for, which is especially helpful with complex numbers.

Imaginary Unit

In complex numbers, the imaginary unit \(i\) is used to represent the square root of -1. It is a fundamental concept because it extends real numbers to include square roots of negative numbers, forming what we call complex numbers.

The property of \(i\) is:

• \(i^2 = -1\)

This property is crucial in simplifying the results of multiplication involving complex numbers. For example, in our step-by-step solution:
When we calculated \((2i)(3i) = 6i^2\), knowing that \(i^2 = -1\) allows us to simplify this to \(6(-1) = -6\). Understanding how to manipulate the imaginary unit is key to working with complex expressions.

Binomial Expansion

Binomial expansion involves multiplying out expressions involving two terms. When dealing with complex numbers, binomial expansion simplifies the multiplication of expressions like \((-1 + 2i)\) and \((-2 + 3i)\).

Once we distribute the terms using the FOIL method, we arrive at an expanded form. For complex numbers, it's particularly important to track both the real and imaginary components. Each pair of terms multiplies to create either a real part or an imaginary part, which are collected after all terms have been expanded.

With binomial expansion, key focus points are:

• Collecting like terms (real parts and imaginary parts separately).
• Applying properties of the imaginary unit (\(i^2 = -1\)) to further simplify terms.

This method not only works in simplifying complex numbers but also broadly applies to algebraic expressions, making it a versatile tool.