Problem 67 - A Graphical Approach to College Algebra

Question

Multiply as indicated. Write each product in standard form. $$i(3-4 i)(3+4 i)$$

Step-by-Step Solution

Answer

The product is \(25i\).

Step 1: Identify the Expression

We want to multiply the complex numbers \( (3 - 4i) \) and \( (3 + 4i) \) as indicated and then multiply the result by \( i \). This requires applying the distributive property and rearranging the result in standard form \( a + bi \).

Step 2: Apply the Difference of Squares

Recognize that \((3 - 4i)(3 + 4i)\) is a difference of squares, given by \((a - b)(a + b) = a^2 - b^2\). Here, \(a = 3\) and \(b = 4i\).

Step 3: Calculate the Conjugate Product

Apply the identity \((3 - 4i)(3 + 4i) = 3^2 - (4i)^2\). This results in \(9 - (16i^2)\). Since \(i^2 = -1\), \((16i^2) = 16(-1) = -16\). Substitute to get \(9 + 16 = 25\).

Step 4: Multiply by i

Now, multiply this real number result, 25, by \(i\): \[i imes 25 = 25i\].

Step 5: Write in Standard Form

The expression is now \(0 + 25i\), which is already in standard form \(a + bi\) with \(a = 0\) and \(b = 25\).

Key Concepts

Multiplication of Complex Numbers

Multiplying complex numbers can be a bit tricky, but it's all about using the distributive property effectively and understanding the nature of imaginary numbers. When you multiply two complex numbers, such as \((a + bi)(c + di)\), you'll follow these steps:

• Use the distributive property to expand the product: \((a + bi)(c + di) = ac + adi + bci + bdi^2\).
• Remember that \(i^2 = -1\). This is crucial because it allows you to simplify terms involving \(i^2\) to real numbers.
• Combine like terms, specifically separating real numbers from imaginary numbers.

The given exercise involved multiplying two conjugates, \((3-4i)\) and \((3+4i)\), and then multiplying the result by \(i\). This is a great opportunity to practice these core multiplication principles.

Standard Form of Complex Numbers

Complex numbers are typically written in what's known as "standard form," which is \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This format makes it easy to identify and work with the real and imaginary parts of a complex number.

After performing operations on complex numbers, such as addition, subtraction, multiplication, or division, it is a standard practice to express the result in this form. In the exercise, the final product after multiplying was given as \(0 + 25i\). This expression is already in standard form:

• The real part, \(a\), is 0.
• The imaginary part, \(b\), is 25.

Writing complex numbers in standard form will help you better analyze and apply them in various mathematical contexts.

Difference of Squares

The difference of squares is a powerful algebraic identity that simplifies expressions and calculations in mathematics. This identity is expressed as \((a - b)(a + b) = a^2 - b^2\). It is particularly useful when working with conjugate pairs of complex numbers.

In the provided exercise, this concept was applied to multiply \((3 - 4i)\) and \((3 + 4i)\). Recognizing them as a difference of squares, the calculation was simplified to \(3^2 - (4i)^2\):

• Calculate \(3^2 = 9\).
• Calculate \((4i)^2 = 16i^2\), which simplifies to \(-16\) because \(i^2 = -1\).
• Simplification leads to \(9 - (-16) = 9 + 16 = 25\).

This clever use of the difference of squares can simplify complex number calculations considerably, highlighting the elegance of algebraic identities.