Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (-2+6 i)(8-i) $$

Step-by-Step Solution

Answer

The expression is \(-10 + 50i\).

Step 1: Expand the Expression Using Distribution

To expand the expression \((-2 + 6i)(8 - i)\), use the distributive property. Multiply each term in the first binomial by each term in the second binomial: \((-2)(8) + (-2)(-i) + (6i)(8) + (6i)(-i)\).

Step 2: Simplify Each Product

Calculate each product from Step 1: 1. \((-2)(8) = -16\)2. \((-2)(-i) = 2i\)3. \((6i)(8) = 48i\)4. \((6i)(-i) = -6i^2 = -6(-1) = 6\) (since \(i^2 = -1\))

Step 3: Combine Like Terms

Combine the real parts and the imaginary parts separately: Real components: \(-16 + 6 = -10\)Imaginary components: \(2i + 48i = 50i\)

Step 4: Write in the Form of \(a + bi\)

Combine the simplified terms from Step 3 into the final form: The expression can be written as \(-10 + 50i\).

Key Concepts

Binomial Expansion

When we talk about binomial expansion in the context of complex numbers, it is simply about multiplying two binomials that may contain both real and imaginary components. This involves a systematic approach to ensure that every term in one binomial is multiplied by every term in the other binomial.

In our example \((-2 + 6i)(8 - i)\), we multiply:

• \((-2)\) by \((8)\) which gives \(-16\)
• \((-2)\) by \((-i)\) which gives \(2i\)
• \((6i)\) by \((8)\) which gives \(48i\)
• \((6i)\) by \((-i)\) which gives \(-6i^2\)

Each of these products is then combined to give a single expression. With binomial expansion, the goal is to carefully follow through all these operations to combine the mixed-products correctly and efficiently.

Distributive Property

The distributive property is an essential arithmetic principle to help simplify expressions involving multiplication.

It dictates that a term can be distributed, or multiplied, across an addition or subtraction inside a parenthesis. In our exercise, it refers to the systematic way we expand our given complex number expression by multiplying each term in the first binomial by each term in the second binomial.

By using the distributive property:

• \((-2 + 6i)\) is distributed over \((8 - i)\)
• This results in the distribution being applied as \((-2)(8) + (-2)(-i) + (6i)(8) + (6i)(-i)\)

The distributive property allows us to handle each combination of terms separately, which makes simplifying the expression much more straightforward.

Imaginary Unit

The imaginary unit, represented as \(i\), is a fundamental concept in complex numbers. It is defined by the equation \(i^2 = -1\). This simple equation leads to a host of interesting properties and behaviors in mathematics.

In our expression, the imaginary unit appears in several multiplications, especially towards the end when we calculate \(-6i^2\). Recognizing that \(i^2 = -1\) allows us to simplify this as \(6\). Understanding that the square of an imaginary unit leads to a negative real number is key when working with complex numbers.

The presence of \(i\) turns real coefficients into imaginary numbers and aids in developing an understanding of how complex numbers work differently from purely real numbers.

Real and Imaginary Components

Complex numbers are composed of both real and imaginary components. The general form for expressing complex numbers is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

In the solution we explored, we identified and combined like terms to sort real from imaginary components. After expanding and simplifying:

• Real components were \(-16 + 6\) resulting in \(-10\)
• Imaginary components were \(2i + 48i\) resulting in \(50i\)

Thus, the complex number can be neatly expressed in the form \(-10 + 50i\). Recognizing and isolating these components helps us present complex numbers in their standard form, which is very useful in mathematics and engineering.