Question

practice assignment 2.4 complex numbers
score: 200/250 answered: 20/25
question 21
divide and simplify:
(-26 + 12i) ÷ (5 + 4i) =
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Explanation:

Step1: Multiply numerator and denominator by the conjugate

Multiply \((-26 + 12i)\) and \((5 + 4i)\) by the conjugate of the denominator \(5-4i\).

$$ \frac{(-26 + 12i)(5 - 4i)}{(5 + 4i)(5 - 4i)} $$

Step2: Expand the numerator and denominator

Use the FOIL method \((a + b)(c + d)=ac+ad+bc+bd\) for the numerator:

$$
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Since \(i^{2}=-1\), we have \(-130 + 104i+60i + 48=-82+164i\).
For the denominator, use \((a + b)(a - b)=a^{2}-b^{2}\), so \((5 + 4i)(5 - 4i)=25-16i^{2}=25 + 16=41\).
The expression becomes \(\frac{-82 + 164i}{41}\).

Step3: Simplify the fraction

Separate the real and imaginary parts: \(\frac{-82}{41}+\frac{164i}{41}=-2 + 4i\)

Answer:

\(-2 + 4i\)