Problem 90 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \((2-3 i)(3+2 i)\)

Step-by-Step Solution

Answer

Combine all parts to get the expression in standard form.

Step 1: Apply the Distributive Property (FOIL)

To simplify the product \((2-3i)(3+2i)\), apply the distributive property. This method is often referred to as FOIL (First, Outer, Inner, Last):1. First: Multiply the first terms: \(2 \times 3 = 6\)2. Outer: Multiply the outer terms: \(2 \times 2i = 4i\)3. Inner: Multiply the inner terms: \(-3i \times 3 = -9i\)4. Last: Multiply the last terms: \(-3i \times 2i = -6i^2\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in mathematics used to multiply one term by a sum of others.
For complex numbers, this property allows us to expand expressions efficiently. When you have a complex number in the form a product like a complex number, you apply the distributive property to reach each component. Essentially, distribute, or "multiply out," each part of one term with each part of the other term.

• Multiply the first term of the first expression by every term of the second expression.
• Repeat this for each term in the first expression.

When applied to complex numbers, it helps simplify expressions that include imaginary units (denoted as 'i'). Understanding this concept is crucial because without it, operations with complex numbers would become cumbersome. It forms the bedrock for many mathematical procedures, including polynomial multiplication.

FOIL Method

The FOIL method is a specialized application of the distributive property used especially for binomials.
It is an acronym that stands for First, Outer, Inner, Last, referring to the order in which you multiply terms. In the context of the expression the FOIL method helps organize these multiplications:

• First: Multiply the first terms of each binomial. For these are the constants (not involving 'i'), producing a real number.
• Outer: Multiply the outer terms of the binomials to get a complex number.
• Inner: Multiply the inner terms, resulting in another complex number.
• Last: Multiply the last terms, often involving 'i' multiplied by 'i'. This step requires converting where we use the fact that

Using the FOIL method effectively breaks down the task into manageable pieces, making it simpler to reach the final solution. Remember that after using FOIL, you'll often need to combine like terms and simplify the expression further, making sure all imaginary terms are correctly calculated.

Standard Form of Complex Numbers

Complex numbers are written as this format clearly separates the real part (a) from the imaginary part (b).
This standard form makes it easier to add, subtract, and compare complex numbers. When you multiply or perform any operation involving complex numbers, ensure your final answer is in this form, simplifying the interpretation and further calculations. After applying the distributive property and FOIL to expand complex expressions, as you simplify:

• Identify and separate the real and imaginary components. Calculate anything involving i-term contributions.
• Combine like terms. Often multiply imaginary numbers like transform
  them, ensure terms are combined accordingly with the real side.
• Express the final result. Make sure it's in the form, clearly showing the real and imaginary parts.

Following these steps ensures your complex number operations are thorough and easy to follow, maintaining clarity throughout the process.