Problem 33 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ (-3+2 i)(-2+i) $$

Step-by-Step Solution

Answer

The expression in standard form is \(4 - 7i\).

Step 1: Use the Distributive Property

The expression \((-3+2i)(-2+i)\) can be expanded using the distributive property, which is also known as the FOIL method for binomials. FOIL stands for First, Outer, Inner, Last, referring to the terms in each binomial. Let's distribute each part:- **First**: Multiply the first terms: \(-3 \times -2 = 6\).- **Outer**: Multiply the outer terms: \(-3 \times i = -3i\).- **Inner**: Multiply the inner terms: \(2i \times -2 = -4i\).- **Last**: Multiply the last terms: \(2i \times i = 2i^2\).

Step 2: Simplify the Expression

Now, sum up all the products obtained:\[6 - 3i - 4i + 2i^2.\]Remember that \(i^2 = -1\), so substitute \(2i^2\) with \(2(-1) = -2\). This simplifies the expression further to:\[6 - 3i - 4i - 2.\]

Step 3: Combine Like Terms

Simplify the expression by combining like terms:- Combine the real numbers: \(6 - 2 = 4\).- Combine the imaginary numbers: \(-3i - 4i = -7i\).Thus, the expression simplifies to:\[4 - 7i.\]

Step 4: Final Answer: Write in Standard Form

The expression in standard form is: \(4 - 7i\). Standard form for a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers.

Key Concepts

Distributive Property

The Distributive Property is a useful tool in algebra that allows you to multiply a single term by each term within a parenthesis. This property is fundamental when working with complex numbers, especially when they are presented in binomial form, like
o(-3 + 2i)(-2 + i)o.

To apply the Distributive Property, you must first understand that each term of the first binomial is multiplied by each term of the second binomial. This process ensures that all possible products are accounted for.

• For instance, the term o(-3)o is multiplied by both o(-2)o and o(i)o.
• Similarly, o(2i)o is multiplied by both o(-2)o and o(i)o.

Using this property simplifies the multiplication process, breaking it down into smaller, more manageable parts. This step-by-step expansion makes it easier to combine like terms in the following steps. By mastering the Distributive Property, you can efficiently handle more complex algebraic operations and solve equations with ease.

FOIL Method

The FOIL Method is a specialized application of the Distributive Property specifically used for multiplying two binomials. FOIL stands for First, Outer, Inner, Last — a sequence that guides you in multiplying the terms. When you encounter an expression like o(-3 + 2i)(-2 + i)o,

you break down each paired multiplication with this method.

• First: Multiply the first terms of each binomial: o(-3)o and o(-2)o.
• Outer: Multiply the outer terms: o(-3)o and o(i)o.
• Inner: Multiply the inner terms: o(2i)o and o(-2)o.
• Last: Multiply the last terms: o(2i)o and o(i)o.

This organized approach not only makes multiplying binomials straightforward but also reduces the chance of missing any terms. After multiplying, you simply combine all the results to simplify the expression. The FOIL Method is especially valuable for students beginning to explore algebraic expressions involving binomials and complex numbers, as it lays out the steps clearly and systematically.

Standard Form of Complex Numbers

Complex numbers are expressed in the form of o(a + bi)o, where o(a)o and o(b)o are real numbers, and o(i)o is the imaginary unit, whose square is o-1o. The Standard Form is essential when communicating mathematical ideas clearly, as it distinguishes between the real and imaginary parts effectively.

After applying the Distributive Property and the FOIL Method to an expression like o(-3 + 2i)(-2 + i)o, you will consolidate like terms to achieve this standard format. The challenge lies in the classification and combination of these terms:

• Combine the real number terms, simplifying them as o6 - 2o which results in o4o.
• Combine the imaginary parts, o-3i - 4io to o-7io.

The final expression o4 - 7io beautifully highlights the simple yet powerful structure of complex numbers: a combination of a tangible real component and the abstract imaginary component. This clarity is crucial, as it provides a standard way to interpret and solve problems involving complex numbers across various fields of mathematics and engineering.