Problem 25 - Precalculus Mathematics for Calculus

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: Distribute the Terms

Use the distributive property to expand the expression \((7 - i)(4 + 2i)\):\[ 7 imes 4 + 7 imes 2i - i imes 4 - i imes 2i \]

Step 2: Simplify the Real and Imaginary Parts

Calculate the results of the individual multiplications determined in Step 1:\(7 imes 4 = 28,\quad7 imes 2i = 14i,\quad-i imes 4 = -4i,\quad-i imes 2i = -2i^2\)

Step 3: Simplify Using \(i^2 = -1\)

Since \(i^2 = -1\), replace \(-2i^2\) with \(2\). Now, substitute into the expansion:\[ 28 + 14i - 4i + 2 \]

Step 4: Combine Like Terms

Combine the real parts and the imaginary parts separately:- Real parts: \(28 + 2 = 30\)- Imaginary parts: \(14i - 4i = 10i\)

Step 5: Write in the form \(a + bi\)

Combine the results from the previous step to express in the desired form:The expression is \(30 + 10i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, crucial when working with expressions involving multiplication and addition. Understanding this property can simplify seemingly complicated problems by breaking them down into manageable steps.
When you apply the distributive property, you mainly allow yourself to multiply each term inside a bracket by a term outside the bracket. For the expression \((7 - i)(4 + 2i)\), this property allows you to distribute each term of the first bracket across every term of the second bracket:

• Multiply \(7\) by each of the terms in \((4 + 2i)\) resulting in \(7 \times 4\) and \(7 \times 2i\).
• Similarly, multiply \(-i\) by each term in \((4 + 2i)\) to get \(-i \times 4\) and \(-i \times 2i\).

This property is immensely useful when dealing with complex numbers, as it simplifies the multiplication of binomials, setting the stage for further simplification.

Imaginary Unit

The imaginary unit, denoted as \(i\), is an essential concept in complex number arithmetic. It is defined by the equation \(i^2 = -1\). This unique property distinguishes imaginary numbers from real numbers and is crucial when performing operations involving complex numbers.
In our original expression, we encountered \(-i \times 2i\), which translates to \(-2i^2\). By knowing that \(i^2 = -1\), you simplify this term as follows:

• Replace \(i^2\) with \(-1\), changing the expression \(-2i^2\) to \(-2 \times (-1)\).
• This further simplifies to just \(+2\), thus impacting the combination of real and imaginary parts.

Understanding the imaginary unit helps you transform expressions effectively, maintaining logical consistency in computations involving complex numbers.

Combining Like Terms

Combining like terms is another essential skill in algebra which helps in simplifying expressions. In the context of complex numbers, like terms refer to the real parts and the imaginary parts of the expression.
For the solution of the expression \((7-i)(4+2i)\) after using the distributive property, you have a set of results that include both real and imaginary parts:

• Real parts: \(28\) and \(2\) from the conversion of \(-2i^2\), resulting in a combined value of \(30\).
• Imaginary parts: \(14i\) and \(-4i\), when added together, result in \(10i\).

By combining like terms, you streamline the expression to form \(a + bi\), where \(a\) is \(30\) and \(b\) is the coefficient \(10\) of the imaginary part. This simplification process is vital, ensuring clarity and uniformity in presenting complex numbers.