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Chapter 3: Problem 34

Perform the indicated operation and express the result as a simplified complex number. \(\frac{2-3 i}{4+3 i}\)

Short Answer

Expert verified
The simplified complex number is \(-\frac{1}{25} - \frac{18}{25}i\).

Step by step solution

01

Multiply by the Conjugate

To simplify the expression \( \frac{2-3i}{4+3i} \), we first multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 4 + 3i \) is \( 4 - 3i \). Thus, we have: \[\frac{(2-3i)(4-3i)}{(4+3i)(4-3i)}\]
02

Calculate the Denominator

Now we calculate the denominator, \((4+3i)(4-3i)\). This is a difference of squares, which simplifies to: \[(4)^2 - (3i)^2 = 16 - 9(-1) = 16 + 9 = 25\]
03

Expand the Numerator

Next, we expand the numerator, \((2-3i)(4-3i)\). Use the distributive property (FOIL method): \[2 \cdot 4 + 2 \cdot (-3i) - 3i \cdot 4 - 3i \cdot (-3i) = 8 - 6i - 12i + 9i^2\] Simplifying this, we get: \[8 - 18i + 9(-1) = 8 - 18i - 9 = -1 - 18i\]
04

Combine the Results

Now that we have simplified both the numerator and denominator, we have: \[\frac{-1-18i}{25}\] This can be separated into real and imaginary components: \[\frac{-1}{25} - \frac{18i}{25}\]
05

Write as a Simplified Complex Number

Finally, express the simplified result in the standard form of a complex number: \(-\frac{1}{25} - \frac{18}{25}i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conjugate of a Complex Number
A complex number is typically in the form of \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The conjugate of a complex number changes the sign of the imaginary part. So, the conjugate of \( a + bi \) is \( a - bi \).

This concept is particularly useful in division involving complex numbers. Multiplying a complex number by its conjugate eliminates the imaginary part, as it results in a real number.
  • In the expression \( \frac{2-3i}{4+3i} \), the conjugate of the denominator \( 4+3i \) is \( 4-3i \).
  • By multiplying the numerator and the denominator by the conjugate, \( (2-3i)(4-3i) \) and \( (4+3i)(4-3i) \) respectively, we simplify the division.

The denominator becomes a real number, making further simplification easier.
Difference of Squares
The difference of squares formula is a vital algebraic tool. It states that \( a^2 - b^2 = (a-b)(a+b) \). This allows for straightforward multiplication of binomials that are a sum and a difference of the same two terms.

In complex numbers, this is critical when dealing with conjugates. For example, multiplying \((4+3i)(4-3i)\) yields a real result thanks to the difference of squares.
  • Set \( a = 4 \) and \( b = 3i \), then \( a^2 - b^2 = 4^2 - (3i)^2 \).
  • Calculating yields \( 16 - 9(-1) \). Since \( i^2 = -1 \), the expression simplifies to \( 16 + 9 = 25 \).
This process cancels out the imaginary components, resulting in a real number denominator, perfect for further calculation.
FOIL Method
The FOIL method is a mnemonic that helps multiply two binomials. It stands for First, Outer, Inner, and Last, referencing the positions of terms in the multiplication.

When applied to complex numbers like \((2-3i)(4-3i)\), it helps break down the expansion process:
  • First: Multiply the first terms: \( 2 \times 4 = 8 \).
  • Outer: Multiply the outer terms: \( 2 \times (-3i) = -6i \).
  • Inner: Multiply the inner terms: \(-3i \times 4 = -12i \).
  • Last: Multiply the last terms: \(-3i \times (-3i) = 9i^2 = 9(-1) = -9 \).

Summing these, you combine like terms \( 8 - 6i - 12i - 9 \) to arrive at a simplified expression \( -1 - 18i \). This makes the multiplication easy and systematic, accurately facilitating complex arithmetic.

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Most popular questions from this chapter

For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form. $$\frac{2 x^{3}+6 x^{2}-11 x-12}{x+4}$$

For the following exercises, determine end behavior of the polynomial function. $$f(x)=2 x^{4}+3 x^{3}-5 x^{2}+7$$

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies inversely as the square of \(x\) and when \(x=1, \quad y=4.\)

Use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$x-2,4 x^{4}-15 x^{2}-4$$

For the following exercises, find all zeros of the polynomial function, noting multiplicities. $$f(x)=(x+3)^{2}(2 x-1)(x+1)^{3}$$
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