In Exercises 21–28, divide and express the result in standard form.


2+3i / 2+i

Verified step by step guidance

1. Identify the given complex division problem: \(\frac{2+3i}{2+i}\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
2. To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \$2+i\( is \)2 - i$.
3. Multiply numerator and denominator by the conjugate: \(\frac{2+3i}{2+i} \times \frac{2 - i}{2 - i} = \frac{(2+3i)(2 - i)}{(2+i)(2 - i)}\).
4. Expand both numerator and denominator using the distributive property (FOIL method): - Numerator: \((2)(2) + (2)(-i) + (3i)(2) + (3i)(-i)\) - Denominator: \((2)(2) + (2)(-i) + (i)(2) + (i)(-i)\).
5. Simplify the expressions by combining like terms and using \(i^2 = -1\), then write the result in the form \(a + bi\), where \(a\) and \(b\) are real numbers.

Key Concepts

Complex Number Division

Dividing complex numbers involves expressing the quotient in a form that separates real and imaginary parts. This is typically done by multiplying numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number, specifically a^2 + b^2, which helps simplify division by removing the imaginary component from the denominator.

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part. Expressing results in this form makes it easier to interpret and use complex numbers in further calculations.