Problem 22 - College Algebra

Question

In Exercises \(21-28,\) divide and express the result in standard form. $$\frac{3}{4+i}$$

Step-by-Step Solution

Answer

In standard form, \(\frac{3}{4 + i}\) is \( \frac{12}{17} - \frac{3}{17}i\).

Step 1: Determine the Conjugate

The conjugate of the complex number \(a + bi\) is \(a – bi\). Therefore, the conjugate of \(4 + i\) is \(4 - i\).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator, \(4 - i\). This yields \(\frac{3 * (4 - i)}{(4 + i) * (4 - i)}\).

Step 3: Simplify the Numerator and Denominator

Apply the distributive property in the numerator to get \(12 - 3i\). In the denominator, apply the difference of squares formula \((a - b)(a + b) = a^2 - b^2\) to get \(4^2 - i^2.\) This simplifies the fraction to \(\frac{12 - 3i}{(16 - (-1))}\), as \(i^2=-1\).

Step 4: Simplify the Answer to Standard Form

Simplify the denominator to get \(17\). Thus, the final result is \(\frac{12 - 3i}{17}\). This fraction can be split into two parts, \( \frac{12}{17} - \frac{3}{17}i\), to express it in the standard form of a complex number.

Key Concepts

Conjugate

The conjugate of a complex number is very straightforward. When you have a complex number in the form of \(a + bi\), its conjugate is \(a - bi\). This operation involves simply changing the sign of the imaginary part. Conjugates are especially useful in division involving complex numbers.
By multiplying a complex number by its conjugate, we can eliminate the imaginary part from the denominator. This step is crucial when dividing complex numbers because it allows us to express the result in standard form without involving the imaginary unit \(i\) in the denominator.

Difference of Squares

The difference of squares is an algebraic identity that is incredibly useful in simplifying expressions. It states that \[(a+b)(a-b) = a^2 - b^2\].
This identity comes in handy when dealing with complex numbers, particularly when the imaginary unit \(i\) is involved. By using this formula, we can eliminate the imaginary part of the denominator in the division of complex numbers. Remember that \(i^2 = -1\), which plays a crucial role in such calculations. By applying this identity, one can transform an expression to a simpler form, making it much more manageable.

Standard Form of Complex Numbers

The standard form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. Expressing a complex number in standard form helps to clearly identify its real and imaginary components.
After performing arithmetic operations on complex numbers, like the division used in this exercise, it's important to end with a result in this form. Writing the answer in standard form makes it easier to interpret and understand. In our exercise, after simplifying the complex fraction, we split it into two parts: the real part \(\frac{12}{17}\) and the imaginary part \(-\frac{3}{17}i\). This separation aligns the result with the standard form of a complex number.

Distributive Property

The distributive property refers to the ability to distribute a multiplication over an addition or subtraction. It states that \[a(b + c) = ab + ac\].
This property is fundamental in algebra and is frequently used in arithmetic involving complex numbers. In the exercise, we applied the distributive property to multiply the numerator by the conjugate of the denominator, resulting in the expression \(12 - 3i\). Using this property allows us to handle expressions that consist of both real and imaginary parts separately and accurately.