Problem 52 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{(1+3 i)(2-4 i)}{(1+2 i)} $$

Step-by-Step Solution

Answer

The expression simplifies to \(3.6 - 5.2i\).

Step 1: Expand the Numerator

First, we need to expand the expression in the numerator: \((1 + 3i)(2 - 4i)\). Use the distributive property (FOIL method):\[(1)(2) + (1)(-4i) + (3i)(2) + (3i)(-4i)\]This simplifies to:\[2 - 4i + 6i - 12i^2\]Since \(i^2 = -1\), replace \(-12i^2\) by \(12\):\[2 - 4i + 6i + 12\]Combine like terms: \(2 + 12 = 14\) and \(-4i + 6i = 2i\)Thus, the expanded numerator is:\(14 + 2i\)

Step 2: Simplify the Expression with the Denominator

Now, take the result from Step 1, \(14 + 2i\), and divide it by \(1 + 2i\).To simplify a complex division, multiply the numerator and the denominator by the conjugate of the denominator.The conjugate of \(1 + 2i\) is \(1 - 2i\). Multiply:\[\frac{(14 + 2i)(1 - 2i)}{(1 + 2i)(1 - 2i)}\]

Step 3: Multiply by the Conjugate

Expand both the numerator and the denominator separately.For the numerator \((14 + 2i)(1 - 2i)\), use FOIL:\[(14)(1) + (14)(-2i) + (2i)(1) + (2i)(-2i)\]This yields:\(14 - 28i + 2i - 4i^2\).Since \(i^2 = -1\), \(-4i^2\) becomes \(4\). Thus:\[14 - 28i + 2i + 4\]Combine like terms: \(14 + 4 = 18\) and \(-28i + 2i = -26i\)Resulting in: \(18 - 26i\).

Step 4: Simplify the Denominator

Expand the denominator \((1 + 2i)(1 - 2i)\) using the difference of squares formula:\[(1)^2 - (2i)^2 = 1 - 4i^2\]Since \(i^2 = -1\), this becomes:\(1 + 4 = 5\).

Step 5: Divide the Expressions

Now, divide the simplified numerator \((18 - 26i)\) by the simplified denominator \(5\):\[\frac{18 - 26i}{5} = \frac{18}{5} - \frac{26}{5}i\].This yields the simplified complex number: \[3.6 - 5.2i\].

Key Concepts

Complex Conjugate

In the world of complex numbers, the **complex conjugate** is a pivotal concept. For any complex number of the form \(a + bi\), its complex conjugate is \(a - bi\). Essentially, you just change the sign of the imaginary part. This operation is useful for rationalizing complex numbers, especially when you're dividing them. When you multiply a complex number by its conjugate, the imaginary parts cancel out, leaving a real number. For example, with \((1 + 2i)\) and its conjugate \((1 - 2i)\), when multiplied together you get

• \((1^2) - (2i)^2\)
• which simplifies to \(1 + 4 = 5\) (because \(i^2 = -1\))

This shows how conjugates help eliminate the imaginary part, making division simpler.

Imaginary Unit

The imaginary unit, denoted by \(i\), is fundamental in complex number arithmetic. It's defined by the relationship \(i^2 = -1\). This definition allows \(i\) to represent the square root of \(-1\), a concept that doesn't exist among real numbers. When performing operations with complex numbers, you'll often find yourself using this property. For instance, in the expression expansion

• \((3i)(-4i) = -12i^2\)
• since \(i^2 = -1\)
• it simplifies to \(12\).

Understanding \(i\) and its properties is crucial, as these operations recur frequently in complex number calculations.

FOIL Method

The FOIL method is a technique used to expand expressions like \((a + bi)(c + di)\). It stands for First, Outer, Inner, Last, referring to a mnemonic for multiplying two binomials. In our problem, we expanded both the numerator and the denominator using this method. Let's break it down with \((1 + 3i)(2 - 4i)\):

• First: \(1 \times 2 = 2\)
• Outer: \(1 \times -4i = -4i\)
• Inner: \(3i \times 2 = 6i\)
• Last: \(3i \times -4i = -12i^2\)

Combining these, we get \(2 - 4i + 6i + 12\). This simplifies further by combining like terms, yielding a cleaner expression \(14 + 2i\). The FOIL method is a reliable strategy for handling complex binomials.

Simplified Complex Number

The goal of evaluating complex number expressions is often to write the result as a simplified complex number. A **simplified complex number** is typically expressed in the form \(a + bi\) where \(a\) and \(b\) are real numbers. To achieve this in our exercise, we divide the simplified expanded forms of the numerator and the denominator. This involves several steps, prominently:

• expanding using FOIL
• applying the complex conjugate to rationalize the denominator
• simplifying the fraction \(\frac{18 - 26i}{5}\)
• which results in: \(3.6 - 5.2i\)

Always aim to present the complex number in this form, as it clearly distinguishes the real part and the imaginary part.