<em>z</em> = 3<em>i</em> / (-1 - <em>i</em> )
<em>z</em> = 3<em>i</em> / (-1 - <em>i</em> ) × (-1 + <em>i</em> ) / (-1 + <em>i</em> )
<em>z</em> = (3<em>i</em> × (-1 + <em>i</em> )) / ((-1)² - <em>i</em> ²)
<em>z</em> = (-3<em>i</em> + 3<em>i</em> ²) / ((-1)² - <em>i</em> ²)
<em>z</em> = (-3 - 3<em>i </em>) / (1 - (-1))
<em>z</em> = (-3 - 3<em>i </em>) / 2
Note that this number lies in the third quadrant of the complex plane, where both Re(<em>z</em>) and Im(<em>z</em>) are negative. But arctan only returns angles between -<em>π</em>/2 and <em>π</em>/2. So we have
arg(<em>z</em>) = arctan((-3/2)/(-3/2)) - <em>π</em>
arg(<em>z</em>) = arctan(1) - <em>π</em>
arg(<em>z</em>) = <em>π</em>/4 - <em>π</em>
arg(<em>z</em>) = -3<em>π</em>/4
where I'm taking arg(<em>z</em>) to have a range of -<em>π</em> < arg(<em>z</em>) ≤ <em>π</em>.
Step-by-step explanation:
1. Find the greatest common factor (GCF)
What is the largest number that divides evenly into 4x^2, -16x^4, and 10x^5?
It is 2.
What is the highest degree of x that divides evenly into 4x^2, -16x^4, and 10x^5?
It is x^2.
Multiply the results above, the GCF = 2x^2
2. Factor out the GCF (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
2x^2(4x^2/2x^2 + -16x^4/2x^2 + 10x^5/2x^2)
3. Simplify each term in parentheses
2x(2-8x^2+5x^3)
Have a nice day :D
