Problem 21 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ (7-i)(4+2 i) $$

Step-by-Step Solution

Answer

The expression evaluates to \( 30 + 10i \).

Step 1: Apply Distributive Property

To evaluate the expression \((7-i)(4+2i)\), start by applying the distributive property: multiply each term in the first binomial by each term in the second binomial: \[7 \cdot 4 + 7 \cdot 2i - i \cdot 4 - i \cdot 2i\]

Step 2: Perform Multiplications

Compute each multiplication:\[7 \times 4 = 28 \7 \times 2i = 14i \-i \times 4 = -4i \-i \times 2i = -2i^2\]

Step 3: Simplify Using Properties of i

Recall that \(i^2 = -1\). Use this knowledge to simplify \(-2i^2\):\[-2i^2 = -2(-1) = 2\]

Step 4: Combine Like Terms

Add up all the real terms and the imaginary terms:Real terms: \(28 + 2 = 30\)Imaginary terms: \(14i - 4i = 10i\)Thus, the expression simplifies to \(30 + 10i\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that allows us to multiply a single term by each term within a set of parentheses. This property is particularly useful when dealing with expressions involving binomials or polynomials.

• To apply the distributive property, take each term from the first group and multiply it by each term in the second group.
• In our exercise, we applied this property to expand \((7-i)(4+2i)\). This means multiplying each term in \((7-i)\) by each term in \((4+2i)\).
• The expanded form is: \(7 imes 4 + 7 imes 2i - i imes 4 - i imes 2i\).

Using the distributive property helps break down complex expressions into simpler parts that are easier to solve. This foundational step lays the groundwork for further simplification of the expression.

Multiplication of Complex Numbers

When multiplying complex numbers, the procedure resembles polynomial multiplication. The rule is to multiply the elements just like ordinary binomials, remembering to handle the imaginary unit \(i\) correctly.

• The imaginary unit \(i\) has a unique property, \(i^2 = -1\). This characteristic must be considered during multiplication.
• In our example: after using the distributive property, the expression expanded to includes products such as \(7 \times 4, 7 \times 2i, -i \times 4,\) and \(-i \times 2i\).
• Calculate each multiplication step-by-step: \(7 \times 4 = 28, 7 \times 2i = 14i, -i \times 4 = -4i,\) and realize \(-i \times 2i = -2i^2\).

Understanding how complex numbers multiply helps manage the imaginary component during calculations.

Simplification

After expanding and multiplying the terms, simplification involves combining similar terms and using properties specific to complex numbers.

• Notice that in our multiplication, \(-2i^2\) becomes positive because \(i^2=-1\), so \(-2(-1)=2\).
• After substitutions, add together all real number components and combine the imaginary components separately.
• In our specific solution, combine \(28 + 2\) to get \(30\) as the real component and \(14i - 4i\) to get \(10i\) as the imaginary component.
• The final answer is written in the standard form of a complex number: \(30 + 10i\).

Simplification compiles our calculation into a concise and easy-to-read final expression. This final step ensures that all arithmetic calculations and imaginary unit properties are correctly applied.