Problem 51 - College Algebra

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: Distribute Terms

First, find the product of the terms in the numerator \((2+i)(4-2i)\). Using the distributive property, we have:\[(2+i)(4-2i) = 2(4) + 2(-2i) + i(4) + i(-2i)\]Calculate each of these terms:- \(2 \cdot 4 = 8\)- \(2 \cdot -2i = -4i\)- \(i \cdot 4 = 4i\)- \(i \cdot -2i = -2i^2\)Since \(i^2 = -1\), \(-2i^2 = 2\).Combine these results: \(8 - 4i + 4i + 2 = 10\).So, the product \((2+i)(4-2i) = 10\).

Step 2: Simplify the Denominator

The denominator is \((1+i)\). We need to simplify this expression.Multiply the numerator and the denominator by the complex conjugate of the denominator. The conjugate of \((1+i)\) is \((1-i)\).The multiplication in the denominator becomes:\[(1+i)(1-i) = 1^2 - (i^2) = 1 + 1 = 2\]This simplifies to \(2\).

Step 3: Multiply by the Conjugate

Now multiply both numerator and denominator by the conjugate \((1-i)\):\[\frac{(10)(1-i)}{(2)}\]Expand the numerator:- \(10 \cdot 1 = 10\)- \(10 \cdot -i = -10i\)So, the expression simplifies to \(10 - 10i\) in the numerator. The denominator is \(2\) from previous steps.

Step 4: Simplify Final Expression

Simplify the expression to become a standard complex number: Divide each term in the numerator by the denominator:\[\frac{10 - 10i}{2} = 5 - 5i\]This gives the simplified form of the complex number \(5 - 5i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It allows you to multiply a sum by multiplying each addend separately and then add those products together. Applying the distributive property is especially useful when working with complex numbers. In the given exercise, we have the expression \((2+i)(4-2i)\). The distributive process involves multiplying each term in the first binomial by each term in the second binomial. Here's how we execute this with our expression:

• Multiply \(2\) by \(4\), which gives you \(8\).
• Multiply \(2\) by \(-2i\), yielding \(-4i\).
• Multiply \(i\) by \(4\), resulting in \(4i\).
• Multiply \(i\) by \(-2i\), which equals \(-2i^2\).Since \(i^2 = -1\), this term becomes \(2\).

Now, simply add these results: \(8 - 4i + 4i + 2\). This simplifies to \(10\) because the imaginary parts cancel each other out.

Complex Conjugate

The complex conjugate is a key tool when simplifying complex fractions. The conjugate of a complex number has the same real part but an opposite sign in the imaginary part. For example, the conjugate of \((1+i)\) is \((1-i)\).In our exercise, we use the conjugate to simplify the expression by eliminating the imaginary part in the denominator. Here's how:

• Multiply both the numerator and the denominator by \((1-i)\), the conjugate of our denominator \((1+i)\).
• When you multiply \((1+i)(1-i)\), you use the difference of squares formula, resulting in \(1^2 - (i^2) = 1 - (-1) = 2\).

This process turns the denominator into a real number, making the expression much easier to simplify.

Simplifying Expressions

Simplifying complex number expressions often involves multiplying the numerator and the denominator by a conjugate or using basic arithmetic operations. From our previous steps, we have reduced the fraction to:\[\frac{(10)(1-i)}{2}\]To simplify the expression, you should expand the numerator by multiplying:

• \(10\) multiplied by \(1\) gives \(10\).
• \(10\) multiplied by \(-i\) gives \(-10i\).

This results in the expression \(10 - 10i\) over the denominator \(2\). Now, divide each term by \(2\) to simplify the overall fraction further into its simplest form.

Standard Form of Complex Numbers

The standard form of complex numbers is expressed as \(a + bi\), where \(a\) represents the real part, and \(bi\) is the imaginary component. This form ensures consistency and makes it easy to identify both parts of a complex number.In the final step of simplifying our expression, \(\frac{10 - 10i}{2}\), we divide each term:

• \(\frac{10}{2} = 5\) is the real part.
• \(\frac{-10i}{2} = -5i\) is the imaginary part.

Thus, the expression \(5 - 5i\) is already neatly arranged in the standard form of a complex number. Presenting your solutions in this format makes them easy to understand and further process.