Problem 47 - Algebra 2 and Trigonometry

Question

In \(46-60,\) write each quotient in \(a+b i\) form. $$ (2+4 i) \div(1-i) $$

Step-by-Step Solution

Answer

The quotient is \(-1 + 3i\).

Step 1: Identify the Conjugate

To divide complex numbers, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number \(a + bi\) is \(a - bi\). For the denominator \(1 - i\), its conjugate is \(1 + i\).

Step 2: Multiply Numerator and Denominator by Conjugate

Multiply both the numerator \((2 + 4i)\) and the denominator \((1 - i)\) by the conjugate of the denominator \((1 + i)\). This gives:\[\frac{(2 + 4i)(1 + i)}{(1 - i)(1 + i)}\]

Step 3: Simplify the Denominator

The denominator becomes:\[(1 - i)(1 + i) = 1^2 - (i)^2 = 1 - (-1) = 2\]

Step 4: Expand the Numerator

Expand the numerator using FOIL method:\[(2 + 4i)(1 + i) = 2 + 2i + 4i + 4i^2\]Simplify to get:\[2 + 6i + 4(-1) = 2 + 6i - 4 = -2 + 6i\]

Step 5: Divide by the Simplified Denominator

Divide each term in the numerator by the denominator:\[-2 + 6i = \frac{-2}{2} + \frac{6i}{2} = -1 + 3i\]

Step 6: Final Answer

Therefore, the quotient in \(a + bi\) form is \(-1 + 3i\).

Key Concepts

Complex Conjugate

Understanding the complex conjugate is essential when dividing complex numbers. The complex conjugate of a number helps us eliminate the imaginary part in the denominator, making division possible. If you have a complex number in the form of \(a + bi\), its conjugate will be \(a - bi\). Essentially, you'll swap the sign of the imaginary part.
For example, the complex conjugate of \(1 - i\) is \(1 + i\). This enables us to transform complex division into a more straightforward multiplication. When you multiply a complex number by its conjugate, the result is a real number, which makes the division easier.

• Example: Conjugate of \((1 - i)\) is \((1 + i)\).
• Multiplying \((1-i)\) by \((1+i)\) results in \(1 + 1 = 2\).

FOIL Method

The FOIL method is a technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, which refers to the position of the terms you multiply.
When dealing with complex numbers, using the FOIL method makes expanding expressions manageable. It's essential to apply this method correctly to get each part of the product.Imagine multiplying the expressions \((2 + 4i)\) and \((1 + i)\):

• First: Multiply the first terms: \(2 \times 1 = 2\).
• Outer: Multiply the outer terms: \(2 \times i = 2i\).
• Inner: Multiply the inner terms: \(4i \times 1 = 4i\).
• Last: Multiply the last terms: \(4i \times i = 4i^2\).

Combine these results as \(2 + 2i + 4i + 4(-1)\), simplifying finally to \(-2 + 6i\). This output is easier to work with in division.

Imaginary Unit

The imaginary unit, often represented as \(i\), is a fundamental concept in complex numbers. It's defined as the square root of \(-1\), meaning that \(i^2 = -1\). Understanding this helps you manage imaginary numbers, crucial in operations involving complex numbers.When you encounter terms like \(4i^2\) in complex multiplication, remember that \(i^2 = -1\). Replace \(i^2\) with \(-1\) to simplify the expression.

• E.g., \(4i^2 = 4 \times (-1) = -4\).

Replacing faulty imaginary components with real numbers using \(i^2\) facilitates the simplification process in operations, such as division and multiplication.

Simplifying Complex Expressions

Simplifying complex expressions allows for expressing them in the standard form \(a + bi\). This process involves breaking down and managing complex operations step-by-step.Let's take \(-2 + 6i\): you set up the expression to divide each real and imaginary part by the value which makes it simpler.

• Divide the entire expression by the real number from the denominator; e.g., \(-2/2 + 6i/2\).
• Then compute: \(-1 + 3i\).

Breaking the numerator down into parts that get divided individually ensures accuracy. With the original quotient \(-1 + 3i\), we've arrived at a form where both the real and imaginary parts are neatly depicted.