Problem 59 - Algebra 2

Question

Simplify. $$ (-3-i)(2-2 i) $$

Step-by-Step Solution

Answer

The simplified form is \(-8 + 4i\).

Step 1: Apply the Distributive Property

To simplify the expression \((-3 - i)(2 - 2i)\), apply the distributive property, also known as the FOIL method in this case involving complex numbers: First, Outside, Inside, Last. Start by multiplying the first terms: \(-3 \times 2 = -6\).

Step 2: Multiply the Outside Terms

Next, multiply the outside terms of the expression: \(-3 \times -2i = 6i\).

Step 3: Multiply the Inside Terms

Multiply the inside terms of the expression: \(-i \times 2 = -2i\).

Step 4: Multiply the Last Terms

Lastly, multiply the last terms of the expression: \(-i \times -2i = 2i^2\). Since \(i^2 = -1\), this simplifies to \(2 \times -1 = -2\).

Step 5: Combine the Results

Combine all the results from steps 1-4: \(-6 + 6i - 2i - 2\).

Step 6: Simplify the Expression

Simplify the expression by combining like terms. Combine the real parts \(-6 - 2 = -8\) and the imaginary parts \(6i - 2i = 4i\). Thus, the simplified expression is \(-8 + 4i\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions. It states that for all numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. This rule is especially useful when dealing with expressions that involve parentheses. By distributing one term over the sum or difference of terms in the parentheses, we can simplify complex expressions. In the given problem, we apply the distributive property in the context of complex numbers.When you have two binomials, such as \((-3-i)(2-2i)\), each term in the first binomial is multiplied by each term in the second binomial. This can ensure that no terms are forgotten and the multiplication is done correctly. This method is sometimes referred to as the FOIL method when applied to two binomials.

FOIL Method

The FOIL method is a specific application of the distributive property, tailored for multiplying two binomials. FOIL stands for First, Outside, Inside, and Last, indicating the order in which you multiply the terms:

• First: Multiply the first terms of each binomial.
• Outside: Multiply the outer terms of the binomials.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms of the binomials.

In our example, \((-3-i)(2-2i)\), apply the FOIL method:- First: \(-3 \times 2 = -6\)- Outside: \(-3 \times -2i = 6i\)- Inside: \(-i \times 2 = -2i\)- Last: \(-i \times -2i = 2i^2\)Each multiplication step follows the standard multiplication rules, helping break down a complex operation into easier, bite-sized steps.

Imaginary Numbers

The concept of imaginary numbers arises when dealing with the square root of negative numbers. The imaginary unit \(i\) is defined such that \(i^2 = -1\). This definition allows us to extend the real number system to solve equations that have no real solutions.When simplifying the expression \((-3-i)(2-2i)\), you encounter terms that involve \(i^2\). Here, \(2i^2\) needs simplification. Since \(i^2 = -1\), it becomes \(2 \times -1 = -2\).This understanding of imaginary units is essential, as it provides a framework for multiplying and simplifying expressions involving complex numbers. Complex numbers have both a real and an imaginary part, represented by \(a+bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Simplification

Simplification is all about combining like terms and making an expression as concise as possible. After applying the FOIL method to \((-3-i)(2-2i)\), you'll have the expression \(-6 + 6i - 2i - 2\).To simplify, combine all the like terms:

• Combine the real parts: \(-6 - 2 = -8\)
• Combine the imaginary parts: \(6i - 2i = 4i\)

As a result, the expression simplifies to \(-8 + 4i\). This step reduces the expression to its simplest form, making it easier to understand and further use in any subsequent calculations. Simplification ensures that any complex arithmetic you do is clear and streamlined.