Problem 70 - A Graphical Approach to College Algebra

Question

Multiply as indicated. Write each product in standard form. $$-5 i(4-3 i)^{2}$$

Step-by-Step Solution

Answer

The product is \(-120 - 35i\).

Step 1: Expand the Expression

First, we need to expand the expression \((4 - 3i)^2\). Apply the formula for squaring a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 4\) and \(b = 3i\), so we calculate: \(4^2 - 2 \times 4 \times 3i + (3i)^2\).

Step 2: Simplify the Squared Terms

Calculate each part: \(4^2 = 16\), \(-2 \times 4 \times 3i = -24i\), \((3i)^2 = 9i^2\). Since \(i^2 = -1\), \(9i^2 = 9(-1) = -9\). Now, combine all components: \(16 - 24i - 9\).

Step 3: Combine Like Terms

Combine the real parts: \(16 - 9 = 7\). This gives us a simplified expression: \(7 - 24i\).

Step 4: Distribute the Scalar Multiple

Multiply \(-5i\) by each term in the expression \((7 - 24i)\).Calculate: \(-5i \times 7 = -35i\) \(-5i \times (-24i) = 120i^2\). Since \(i^2 = -1\), substitute to get \(120i^2 = 120(-1) = -120\).

Step 5: Final Solution

Combine the terms from the multiplication: \(-120 - 35i\). This is the product in standard form.

Key Concepts

Imaginary unit

The imaginary unit, denoted as \(i\), is a fundamental concept in mathematics, particularly when dealing with complex numbers. The imaginary unit is defined by the property that \(i^2 = -1\). This gives rise to numbers that are called "imaginary" because they do not exist on the traditional number line. Instead, they exist on an "imaginary" number line perpendicular to the real number line:

• Imaginary numbers are written in the form \(bi\), where \(b\) is a real number.
• An example of an imaginary number is \(3i\), which equals \(3 \times i\).

When combined with real numbers, imaginary numbers form complex numbers. It is crucial to understand that \(i^2 = -1\) because this property is used in operations such as squaring binomials and multiplying complex numbers, as shown in our exercise.

Binomial expansion

The binomial expansion is a process used to expand expressions that are raised to a power. A binomial is a polynomial with two terms, such as \((a + b)\). When expanding a binomial like \((a - b)^2\), we use the formula:

• \((a - b)^2 = a^2 - 2ab + b^2\)

In the exercise provided, \((4 - 3i)^2\) is expanded using this formula. This yields:

• First, calculate \(4^2 = 16\).
• Then, find \(-2 \times 4 \times 3i = -24i\).
• Finally, compute \((3i)^2 = 9i^2\). Since \(i^2 = -1\), this becomes \(-9\).

Combine these results to get:

• \(16 - 24i - 9\) which simplifies to \(7 - 24i\).

The binomial expansion is essential for breaking down and solving complex expressions, enabling us to combine and simplify terms effectively.

Complex multiplication

Complex multiplication involves multiplying complex numbers, which can be in the form \((a + bi)\). The process is similar to the distributive property but includes handling \(i^2 = -1\).

• When you multiply two complex numbers, ensure each component is multiplied, including both the real and imaginary parts.
• An essential step is remembering that terms like \(i \times i = i^2\), which equals \(-1\).

In the given exercise, we see an example of this concept:

• The term \(-5i\) is multiplied by \((7 - 24i)\).
• Calculate \(-5i \times 7\) to get \(-35i\).
• Then, \(-5i \times (-24i) = 120i^2\), which becomes \(-120\) after substituting \(i^2 = -1\).

Combine these results to reach the solution: \(-120 - 35i\). This illustrates how to manage complex numbers involving imaginary units and achieve a result in standard form.

Standard form of complex numbers

The standard form of complex numbers is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

• The term \(a\) represents the real part of the complex number.
• The term \(bi\) represents the imaginary part.

This form makes it easy to perform arithmetic operations and understand the structure of complex numbers. In the context of our exercise:

• After multiplication and simplification, we acquire the result \(-120 - 35i\).
• Here, \(-120\) is the real part, and \(-35i\) is the imaginary part.

By expressing numbers in standard form, the real and imaginary parts are clearly distinguished. This form not only simplifies arithmetic operations but also provides a straightforward means of interpreting complex numbers in geometrical terms, where the complex plane is utilized. The real part lies along the horizontal axis, and the imaginary part lies along the vertical axis.