Problem 10 - Elementary and Intermediate Algebra

Question

Fill in the blanks. Factor each numerator. Then remove the factor common to the numerator and denominator. Write the result in the form \(a+b i\) a. \(\frac{3+6 i}{3}\) b. \(\frac{15+25 i}{10}\)

Step-by-Step Solution

Answer

a. \(1 + 2i\); b. \(\frac{3}{2} + \frac{5}{2}i\)

Step 1: Factor the Numerator (Part a)

For the given expression \( \frac{3 + 6i}{3} \), first factor out the common factor in the numerator. The common factor of the terms in the numerator \(3 + 6i\) is 3. Thus, we can write:\[ 3(1 + 2i) \]

Step 2: Cancel the Common Factor (Part a)

Now, cancel the common factor of 3 in both the numerator \(3(1 + 2i)\) and the denominator 3. The expression becomes:\[ 1 + 2i \]

Step 3: Simplified Form (Part a)

After canceling the common factor, the expression \(\frac{3 + 6i}{3}\) simplifies to \(1 + 2i\), which is in the standard form \(a + bi\).

Step 4: Factor the Numerator (Part b)

For the expression \( \frac{15 + 25i}{10} \), factor out the common factor in the numerator. The common factor is 5, producing:\[ 5(3 + 5i) \]

Step 5: Cancel the Common Factor (Part b)

Cancel the common factor of 5 in both the numerator \(5(3 + 5i)\) and the denominator 10, which simplifies to:\[ \frac{3 + 5i}{2} \]

Step 6: Simplified Form (Part b)

The expression \(\frac{15 + 25i}{10}\) simplifies to \(\frac{3}{2} + \frac{5}{2}i\) after canceling out the common factor.

Key Concepts

Understanding Factorization

Factorization is the process of breaking down an expression into a product of simpler factors. For example, consider the expression \(3 + 6i\). Here, you identify the greatest common factor that can divide all terms in the expression. In this case, the number 3 is common in both terms, so you factor it out:

• The numerator \(3 + 6i\) can be rewritten as \(3(1 + 2i)\).

After factorization, you can simplify expressions by cancelling out any common factors present in both the numerator and the denominator.

• In example a, this process allows \(\frac{3 + 6i}{3}\) to become \(1 + 2i\) after cancelling the 3.
• For example b, \(\frac{15 + 25i}{10}\) simplifies to \(\frac{3}{2} + \frac{5}{2}i\) by cancelling the common factor 5.

Practice makes perfect when it comes to factorization. The key is correctly identifying the common factor before proceeding with the simplification steps.

Rational Expressions with Complex Components

A rational expression is a fraction where the numerator and the denominator are polynomials. When dealing with complex numbers within these rational expressions, you process them similarly to normal rational expressions.

• The goal is to simplify by identifying and removing any common factors from both parts.
• With complex numbers, the key is handling the numbers carefully due to the imaginary components.

In exercise b, the rational expression \(\frac{15 + 25i}{10}\) initially looks complex. However, by factoring out the common factor in the numerator and cancelling it with the denominator, the expression gradually becomes more manageable, leading to a simpler form \(\frac{3}{2} + \frac{5}{2}i\).

• This form is helpful because it breaks the numbers into real and imaginary parts, making operations like addition or multiplication easier.

Understanding rational expressions that involve complex numbers is essential for simplifying and performing operations on more intricate algebraic problems.

The Imaginary Unit and Its Role

The imaginary unit, represented as \(i\), is a fundamental concept in complex numbers. It is defined by the property that \(i^2 = -1\). This allows for the extension of real numbers to form complex numbers, where \(a + bi\) represents the combination of real number \(a\) and imaginary part \(bi\).

• "Imaginary" does not mean unreal—it’s a different dimension of numbers.

In our examples, the expressions include terms with \(i\), indicating their place in the complex realm. Simplifying expressions like \(1 + 2i\) or \(\frac{3}{2} + \frac{5}{2}i\) involves dealing with these imaginary units straightforwardly.

• Each term \(bi\) signifies a unique direction on the complex plane, which adds depth to mathematical operations.

Grasping the concept of the imaginary unit is vital, as it opens math to methods and results beyond the limitations of the real number system.