Problem 6 - Advanced Engineering Mathematics

Question

In Problems 1-26, write the given number in the form \(a+i b\). $$ 3(4-i)-3(5+2 i) $$

Step-by-Step Solution

Answer

The number is \(-3 - 9i\).

Step 1: Distribute the factors

First, apply the distributive property to both expressions inside the parentheses. Multiply 3 with each term inside the first parenthesis: \( 3 \times 4 - 3 \times i = 12 - 3i \). Multiply -3 with each term inside the second parenthesis: \( -3 \times 5 - 3 \times 2i = -15 - 6i \).

Step 2: Combine like terms

Now, add the results from Step 1 together. Combine the real parts: \( 12 - 15 = -3 \). Combine the imaginary parts: \( -3i - 6i = -9i \).

Step 3: Write in the form a + ib

Combine the results from Step 2 to write the expression in the standard form for a complex number, which is \(a + ib\).So, the final result is: \(-3 - 9i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term within a parenthesis. It states that for any numbers or expressions \(a\), \(b\), and \(c\), the expression \(a(b + c)\) can be expanded to \(ab + ac\). This rule ensures each term within the parentheses is affected by the outside multiplier. In our example, we used the distributive property to transform \(3(4-i)\) into \(12 -3i\), and \(-3(5+2i)\) into \(-15 - 6i\). By breaking down expressions with parentheses, we use multiplication and follow the sequence carefully to ensure no term is missed. This initial distribution sets the stage for further simplification, like combining like terms.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have identical variables. In the context of complex numbers, this means identifying and combining real parts and imaginary parts separately. After applying the distributive property, we often end up with several terms, some of which share similarities. For example, in the expression \(12 - 3i - 15 - 6i\), we combine the real coefficients \(12\) and \(-15\) to get \(-3\), and the imaginary coefficients \(-3i\) and \(-6i\) to get \(-9i\).

• The real parts are combined to simplify the expression further.
• The imaginary parts are also combined similarly.

This process helps in writing the expression in a cleaner form that is easy to interpret, especially when working with complex numbers.

Complex Number Operations

Complex numbers follow specific rules when subjected to various operations like addition, subtraction, and multiplication. They consist of a real part and an imaginary part, usually expressed as \(a + ib\), where \(a\) is the real part and \(b\) is the imaginary part. The operations with complex numbers are straightforward when you understand that real and imaginary parts must be handled separately.
For addition and subtraction, add or subtract the real parts and the imaginary parts separately. In the example \(-3 - 9i\), \(-3\) is the resultant real part from combining \(12\) and \(-15\), while \(-9i\) is the resultant imaginary part from combining \(-3i\) and \(-6i\).

• The resulting number after these operations is a new complex number in the form \(a + ib\).

This systematic approach ensures that we maintain the correct format while simplifying expressions that involve complex numbers.