Problem 10 - Intermediate Algebra

Question

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

Step 1: Factor the Numerator of (a)

The numerator of the first expression is \(3 + 6i\). Notice that both terms have a common factor of 3. So, we factor out 3: \[ 3(1 + 2i) \] Thus, the expression can be rewritten as: \[ \frac{3(1 + 2i)}{3} \]

Step 2: Simplify (a) by Removing Common Factors

Now that we have factored the numerator, we can remove the common factor of 3 from both the numerator and the denominator: \[ \frac{3(1 + 2i)}{3} = 1 + 2i \]The result in the form \(a + bi\) is \(1 + 2i\).

Step 3: Factor the Numerator of (b)

The numerator of the second expression is \(15 + 25i\). Here, both terms are divisible by 5. We factor out 5: \[ 5(3 + 5i) \] Thus, the expression becomes: \[ \frac{5(3 + 5i)}{10} \]

Step 4: Simplify (b) by Removing Common Factors

The denominator 10 can be rewritten as \(2 \times 5\). Now, we can cancel the common factor of 5 from both the numerator and the denominator: \[ \frac{5(3 + 5i)}{10} = \frac{3 + 5i}{2} \]The result in the form \(a + bi\) is \(\frac{3}{2} + \frac{5}{2}i\).

Key Concepts

Factoring

Factoring is a mathematical process where we express a number or a term as a product of its factors. In complex numbers, we aim to identify a common factor from the terms involved. The goal is to simplify expressions, making them easier to work with or solve.

• For instance, with the expression \(3 + 6i\), both 3 and 6i have a common factor of 3.
• Similarly, in \(15 + 25i\), both terms are divisible by 5.

By factoring these numbers, we can rewrite the expressions as \(3(1 + 2i)\) and \(5(3 + 5i)\) respectively. This step sets the stage for further simplification.

Simplifying Expressions

Once you've factored an expression, the next step is to simplify it. In mathematics, simplifying means reducing complexity.
To do this, especially in complex numbers, you'll want to remove common factors from both the numerator and the denominator.

• In \(\frac{3(1 + 2i)}{3}\), the common factor, 3, can be canceled out.
• This leaves us with \(1 + 2i\).
• In \(\frac{5(3 + 5i)}{10}\), the 5 in the numerator cancels with one 5 from the denominator.

After this cancellation, the expression simplifies to \(\frac{3 + 5i}{2}\).
Through simplification, expressions are left in a tidy and more usable form.

Numerator and Denominator

The numerator and denominator are parts of a fraction. The numerator is the top part, and the denominator is the bottom part.

• For fractions like \(\frac{3 + 6i}{3}\), \(3 + 6i\) is the numerator and 3 is the denominator.
• Manipulating these parts is key to simplifying fractions.

Breaking down the numerator helps reveal any factors that can also divide the denominator. Removing these common factors provides a simpler version of the fraction.
This simplification makes calculations and analyses easier, especially when dealing with complex numbers.

Complex Form a+bi

Complex numbers can be expressed in the form \(a + bi\). In this format, "a" is the real part and "bi" is the imaginary part.

• When we simplify the expression \(\frac{3 + 6i}{3}\) into \(1 + 2i\), it fits neatly into the \(a + bi\) structure.
• Likewise, simplifying \(\frac{15 + 25i}{10}\) gives us \(\frac{3}{2} + \frac{5}{2}i\), which also aligns with this standard form.

Using the \(a + bi\) form helps communicate and work with complex numbers effectively. It allows for easier further calculations and is a universal structure in mathematics involving complex numbers.