Problem 51 - Precalculus Student Solutions Manual 5th

Question

Find each product. Write the answer in standard form. $$(2+i)(3-2 i)$$

Step-by-Step Solution

Answer

8 - i

Step 1: Apply the Distributive Property

To find the product \( (2+i)(3-2i) \), employ the distributive property, also known as the FOIL method (First, Outer, Inner, Last). You will multiply each term in the first binomial by each term in the second binomial.

Step 2: Multiply the First Terms

Multiply the first terms of each binomial: \( 2 \times 3 = 6 \).

Step 3: Multiply the Outer Terms

Multiply the outer terms of each binomial: \( 2 \times -2i = -4i \).

Step 4: Multiply the Inner Terms

Multiply the inner terms of each binomial: \( i \times 3 = 3i \).

Step 5: Multiply the Last Terms

Multiply the last terms of each binomial: \( i \times -2i = -2i^2 \). Recall that \( i^2 = -1 \), so \( -2i^2 = -2(-1) = 2 \).

Step 6: Combine Like Terms

Combine all the results from the previous steps: \( 6 - 4i + 3i + 2 \.\) Combine the real numbers (6 and 2), and combine the imaginary parts (-4i and 3i): \( 8 - i \).

Key Concepts

Distributive Property

When working with expressions like \((2+i)(3-2i)\), you use the distributive property to multiply each term in the first binomial by every term in the second binomial. This is essential to ensure all parts of each polynomial are covered.
This property is sometimes remembered by the acronym FOIL, which stands for First, Outer, Inner, Last.
For instance:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms.

The distributive property ensures that all components are included in the final product.

FOIL Method

The FOIL method is a specific application of the distributive property used for multiplying two binomials. Here's a deeper look into each step:

• First: Multiply the first terms from each binomial, \(2 \cdot 3 = 6\).
• Outer: Multiply the outer terms, \(2 \cdot -2i = -4i\).
• Inner: Multiply the inner terms, \(i \cdot 3 = 3i\).
• Last: Multiply the last terms, \(i \cdot -2i\). Remember that \(i^2 = -1\), so \(-2i^2 = -2(-1) = 2\).

Finally, combine all the products: \(6 - 4i + 3i + 2\). By simplifying like terms, we get the standard form: \(8 - i\).

Imaginary Numbers

Imaginary numbers revolve around the concept of \(i\), which is defined as \(i = \sqrt{-1}\). In practical terms, \(i^2 = -1\). This unique property allows for handling roots of negative numbers.

In our example, multiplying \(i\) values also includes their unique interaction:

• \(i \cdot i = i^2\)
• Since \(i^2 \) is \(-1\), multiplying further by constants adjusts to \(–1\), turning negative forms positive when combined, as in \(-2i^2 = 2\).

Understanding imaginary numbers is crucial for complex multiplication, ensuring that square roots of negatives are managed effectively.

Standard Form in Complex Numbers

The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. To convert a product to this form:

• Combine real parts.
• Combine imaginary parts.

For \((2+i)(3-2i)\), after applying the distributive/FOIL method, and simplifying, the result was \(8 - i\).
Here:

• Real part: 8
• Imaginary part: -i

Having complex numbers in this format makes them easier to work with in mathematics, especially for addition, subtraction, and further multiplication.