Problem 19 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ 4(-1+2 i) $$

Step-by-Step Solution

Answer

The expression evaluates to \(-4 + 8i\).

Step 1: Identify the Expression to Distribute

The expression given is \(4(-1 + 2i)\). We need to distribute the coefficient 4 among each term inside the parentheses.

Step 2: Distribute the Coefficient

Multiply each term in the parentheses by 4: - First, multiply \(-1\) by 4, which equals \(-4\).- Then, multiply \(2i\) by 4, which equals \(8i\).

Step 3: Combine the Results

Combine the results of both multiplications to get the final expression: \(-4 + 8i\).

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

The expression \(-4 + 8i\) is already in the form of \(a + bi\), where \(a = -4\) and \(b = 8\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept used to simplify expressions by spreading or "distributing" a multiplication operation over a sum or difference inside parentheses. It states that multiplying a sum or a difference by a number is the same as doing each multiplication separately, then adding or subtracting the results. This property can be expressed as: \(a(b + c) = ab + ac\).

In the context of complex numbers, the distributive property is particularly useful. When a real number is multiplied by a complex number expression, such as \(4(-1 + 2i)\), you apply the distributive property to multiply each term inside the parentheses by the number outside.

• First, multiply \(-1\) by \(4\), resulting in \(-4\).
• Next, multiply \(2i\) by \(4\), resulting in \(8i\).

By using the distributive property, the expression is simplified to \(-4 + 8i\). This method helps maintain accuracy and simplifies calculations involving complex numbers.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a key element in complex numbers. It represents the square root of \(-1\), which is not a real number but is used to extend the real number system to include imaginary numbers. In mathematical expressions, \(i\) is used to denote imaginary parts. This leads to complex numbers having a form of \(a + bi\), where \(a\) and \(b\) are real numbers.

• \(i^2 = -1\), which means the square of the imaginary unit is \(-1\).
• Operations with \(i\) follow standard arithmetic rules, but keep in mind its property when squaring or simplifying expressions.

In the example \(4(-1 + 2i)\), \(i\) is part of the term inside the parentheses. After using the distributive property, you multiply \(2i\) by \(4\), resulting in \(8i\). Understanding the role of \(i\) in such expressions is critical for performing operations with complex numbers.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, operators (like +, −, ×, ÷), and grouping symbols. In complex numbers, algebraic expressions often include both real and imaginary components, resulting in expressions of the form \(a + bi\).

When dealing with expressions such as \(4(-1 + 2i)\), understanding how to manipulate and simplify both the real and imaginary parts is essential. This often involves recognizing how to apply properties like the distributive property and understanding the imaginary unit.

• Real part: In the expression \(-4 + 8i\), \(-4\) is the real part.
• Imaginary part: \(8i\) is the imaginary part.

By simplifying complex algebraic expressions, you can express them in their standard form \(a + bi\), making it easier to perform further operations or solve equations involving complex numbers.