Problem 76 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(10+2 i)(-2-i)$$

Step-by-Step Solution

Answer

The product is \(-18 - 14i\).

Step 1: Distribute the First Term

Use the distributive property to multiply the first term from the first binomial by each term in the second binomial: \[10 \times (-2) + 10 \times (-i) = -20 - 10i\].

Step 2: Distribute the Second Term

Now, distribute the second term from the first binomial over each term in the second binomial:\[2i \times (-2) + 2i \times (-i) = -4i - 2i^2\].

Step 3: Simplify and Combine Like Terms

Combine all the terms obtained from the previous steps: \[-20 - 10i - 4i - 2i^2\]. Since \(i^2 = -1\), substitute \(-2i^2\) with \(+2\), which simplifies the expression to: \[-20 - 14i + 2\].

Step 4: Final Simplification

Combine the real parts and the imaginary parts: \[-20 + 2 - 14i = -18 - 14i\]. Thus the product in the standard form of a complex number is \(-18 - 14i\).

Key Concepts

Algebraic Operations

Algebraic operations are fundamental in manipulating and solving equations and expressions. Within the realm of complex numbers, algebraic operations such as addition, subtraction, multiplication, and division function similarly to how they would with real numbers, with a few additional rules.
In our exercise, we perform multiplication between two complex numbers, each represented by a binomial form (e.g., \(a+bi\)). Multiplication involves several algebraic steps: distributing terms, multiplying the real part separately, and simplifying with the imaginary part. It’s important to apply these operations while accounting for the presence of the imaginary unit, denoted as \(i\). For example, when multiplying \((-2-i)\) by a complex number, you handle each term in the binomial separately before combining results.
By mastering these algebraic operations, students can solve complex equations, thereby deepening their understanding of complex number systems.

Distributive Property

The distributive property is a handy tool in algebra used to simplify expressions by breaking them down. It's particularly useful when multiplying complex numbers, as shown in our problem. The property states that \(a(b + c) = ab + ac\).
Let's use the distributive property to multiply the binomials \((10+2i)(-2-i)\). First, multiply the term \(10\) across the two terms in the second binomial: \(10 imes (-2)\) and \(10 imes (-i)\). Next, distribute \(2i\) to each term: \(2i imes (-2)\), \(2i imes (-i)\). Distribute each term, then combine to simplify your expression.
Applying the distributive property simplifies what could otherwise be a complicated multiplication, allowing handling of each term separately while ensuring no mistakes.

Imaginary Unit

The imaginary unit, \(i\), is the cornerstone of complex numbers. Defined as \(i = \sqrt{-1}\), it introduces an entirely new dimension to algebraic operations. In standard algebra, squaring a number results in a positive value, but with the imaginary unit, \(i^2 = -1\).
Understanding the roles of \(i\) is crucial in manipulating complex numbers. For instance, in our exercise, the term \(2i imes (-i)\) leads to \(-2i^2\), which can be simplified using the fact that \(i^2 = -1\) to \(+2\). This unique property of \(i\) transforms otherwise complex expressions into more manageable forms.
Thus, this small symbol opens up vast opportunities to explore mathematical problems beyond the realm of real numbers.

Standard Form of Complex Numbers

Complex numbers are usually expressed in their standard form, \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. This notation clearly separates the two components of the number, aiding in their manipulation and understanding.
When solving multiplication problems in complex numbers, like in our exercise, it’s important to express the final answer in standard form. After performing all operations, you recombine terms to fit the \(a + bi\) format. In our case, the equation resolves to \(-18 - 14i\), where \(-18\) is the real part and \(-14i\) is the imaginary part.
Formulating the solution in standard form is key for clear communication of results, allowing one to quickly ascertain the real and imaginary components of any complex number.