Problem 32 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ (-2+i)(1-2 i) $$

Step-by-Step Solution

Answer

The expression is 2 + 5i.

Step 1: Apply the Distributive Property

We start by distributing each term in the first binomial (-2 + i) to each term in the second binomial (1 - 2i). This involves four multiplications: 1. (-2) imes 1 2. (-2) imes (-2i) 3. i imes 1 4. i imes (-2i).

Step 2: Perform Each Multiplication

Let's calculate each multiplication from the previous step: 1. (-2) imes 1 = -2 2. (-2) imes (-2i) = 4i 3. i imes 1 = i 4. i imes (-2i) = -2i^2. Since i^2 = -1, we have -2i^2 = -2(-1) = 2.

Step 3: Combine Like Terms

Now, we need to add all the terms obtained from the multiplication: a = -2 b = 4i + i = 5i c = 2 Thus, the expression is (-2) + 5i + 2 = 2 + 5i.

Step 4: Write the Expression in Standard Form

The standard form of a complex number is a + bi. In this case, we have obtained: 2 + 5i. Therefore, the expression (-2 + i)(1 - 2i) can be expressed in standard form as 2 + 5i.

Key Concepts

Distributive Property

The distributive property is a fundamental arithmetic rule that applies to multiplication over addition or subtraction. In simple terms, when you have an expression such as \((-2 + i)(1 - 2i)\), you apply the distributive property by multiplying each term in one bracket by every term in the other bracket. This results in:

• First, \(-2 \times 1\) = -2.
• Second, \(-2 \times (-2i)\) = 4i.
• Third, \(i \times 1\) = i.
• And finally, \(i \times (-2i)\) = -2i^2.

By multiplying each number this way, we ensure that no terms are left out. This principle is key to simplifying complex number expressions or equations with multiple terms. After using the distributive property, we combine all results to simplify the expression.

Standard Form

The standard form of a complex number is a way of expressing the number clearly. It is written in the format \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. For the expression \((-2 + i)(1 - 2i)\), applying the solution steps gives us:

• The constant terms sum up to give the real part, \((-2) + 2\), which simplifies to 2.
• The imaginary terms sum up to give the imaginary part, \(4i + i\), which simplifies to \(5i\).

Combining these, the expression becomes \(2 + 5i\). Standard form makes it easy to identify and work with the individual parts of a complex number, which is important in performing operations like addition and subtraction.

Imaginary Unit

The imaginary unit is an essential component of complex numbers, represented by the letter \(i\). It is understood as the square root of -1, that is \(i^2 = -1\). This concept might seem confusing at first, but it is valuable in various math and engineering fields. In the solution, when we calculated \(i \times (-2i)\), we needed to replace \(i^2\) with -1:

• Starting with \(-2i^2\), we substitute by recognizing \(i^2 = -1\).
• This gives us \(-2(-1)\), resulting in 2.

By understanding how the imaginary unit functions, you can correctly handle operations that involve complex numbers. Recognizing \(i^2\)'s value is crucial to simplifying and solving expressions efficiently.