Problem 51 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The simplified complex number is \(5 - 5i\).

Step 1: Expand the Numerator

First, we need to expand the numerator \((2 + i)(4 - 2i)\) by using the distributive property (also known as the FOIL method):\[(2+i)(4-2i) = 2 \cdot 4 + 2 \cdot (-2i) + i \cdot 4 + i \cdot (-2i)\]This results in:\[= 8 - 4i + 4i - 2i^2\]Since \(i^2 = -1\), our expression simplifies to:\[8 - 2(-1) = 8 + 2 = 10\]So the expanded form of the numerator is \(10\).

Step 2: Address the Denominator

The denominator of the expression is \((1+i)\). To simplify the expression, we can multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \((1+i)\) is \((1-i)\).

Step 3: Multiply by the Conjugate

Multiply both the numerator and the denominator by \((1-i)\) to simplify the division:\[\frac{10 \cdot (1-i)}{(1+i)(1-i)}\]The denominator becomes:\[(1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2\]

Step 4: Simplify Resulting Expression

Now, simplify the numerator:\[10 \cdot (1-i) = 10 - 10i\]Thus, the full expression is:\[\frac{10 - 10i}{2}\]Now divide each term by 2:\[= \frac{10}{2} - \frac{10i}{2} = 5 - 5i\]

Step 5: Final Simplified Complex Number

The simplified complex number is:\[5 - 5i\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept frequently used in algebra, including complex number arithmetic. It allows us to expand expressions neatly. Think of it as distributing or spreading multiplication over addition or subtraction inside parentheses. When we look into expressions like \((a+b)(c+d)\), the distributive property lets us expand this as:\[a \/cdot c + a \/cdot d + b \/cdot c + b \/cdot d\] In our example, \((2+i)(4-2i)\), we utilize this property to separate and calculate each part:- Multiply 2 by 4- Multiply 2 by \(-2i\)- Multiply \(i\) by 4- Multiply \(i\) by \(-2i\)After combining all these terms, we see how every element from the first expression is multiplied by every element from the second expression. This process simplifies our expressions step by step.

Complex Conjugate

Understanding the complex conjugate is vital when dealing with division in complex numbers. A complex number, such as \(a+bi\), has a complex conjugate \(a-bi\). You simply flip the sign between the real and imaginary components. Why is this important? Multiplying a complex number by its conjugate results in a real number. This can dramatically simplify certain calculations, especially division.For instance, in our problem involving \(\frac{10}{(1+i)}\), we choose the complex conjugate \(1-i\) to clear the imaginary component from the denominator. This is a common strategy to make the math much cleaner and ensure we keep expressions straightforward. Remember, using the complex conjugate effectively requires changes to both the numerator and denominator, maintaining the equation's balance.

Simplification

Simplifying complex numbers often involves combining like terms and removing unnecessary components. This is where simplifying really shines! After our initial multiplication and application of the distributive property, we consistently seek simplification.First, notice terms like \(-2i^2\). Recall that owing to the unique behavior of the imaginary unit \(i\), where \(i^2 = -1\), you can substitute this to further simplify. Thus, \(-2i^2\) becomes \(2\), and so our expression condenses significantly. By acknowledging such relationships and pursuing simplification at each step, from multiplying to reducing fractions, your final expression gets cleaner, as seen: \[5-5i\].Mastering the art of simplification ensures accuracy and elegance in your final results.

FOIL Method

The FOIL Method is an extension of the distributive property, specifically adapted for multiplying binomials. The acronym FOIL stands for:

• First
• Outside
• Inside
• Last

When using FOIL on \((2 + i)(4 - 2i)\), each letter in FOIL signifies which terms to multiply:- First: Multiply first terms, \(2 \cdot 4\)- Outside: Multiply outside terms, \(2 \cdot -2i\)- Inside: Multiply inside terms, \(i \cdot 4\)- Last: Multiply last terms, \(i \cdot -2i\)By addressing these pairs systematically, it's easy to handle complex multiplication without missing any crucial parts. Just follow the sequence and add the products together. This technique keeps you organized and is particularly handy for complex numbers, where arranging and dealing with terms efficiently become crucial.