Elementary English & Math Education Tips

I need with my math homework pre-calculus ?

Ok heres the scoop, I have an important assignment due tomorow in my pre-calculus (elementary functions) math class, it consists of 40 or so questions, and i have completed all but three. These three have stumped me and i think it is because i missed the days of class that it was being taught. Anyways I would greatly appreciate any help from anyone who could help me just get through these problems. Thanks to you guys in advance for any help you provide!

Q1)
Standard form is “z = a + bi”

Given z:
= (3-i)²(3+i) / (2-i)
= (3-i)(3-i)(3+i)(2+i) / (2-i)(2+i)
= (3-i)(9-3i+3i+1)(2+i) / (4-2i+2i+1)
= 10(3-i)(2+i)/5
= 10(6-2i+3i+1)/5
= 10(7+i)/5
= 14 + 2i

Q2)
To find the polynomial use The Factor Theorem and The Complex Conjugate Root Theorem to obtain the factors:
f(x) = (x+2)(x+2)(x-1+3i)(x-1-3i)
f(x) = (x²+4x+4)(x²-x+3ix-x+1-3i-3ix+3i+9)
f(x) = (x²+4x+4)(x²-2x+10)

Which can be expanded to:
f(x) = x^4 + 2x^3 + 6x^2 + 32x + 40

Q3)
If (2-3i) is a root of the function, then (2+3i) is also a root.

Therefore, the function has factors (x-2+3i) and (x-2-3i).

Which when multiplied give:
(x² – 2x + 3ix – 2x + 4 – 6i – 3ix + 6i + 9)
= (x² – 4x + 13)

Given that:
(x² – 4x + 13)(x² + 3x + 3) = x⁴- 4x³ + 16x² – 12x + 39

Therefore as the fourth order polynomial divides evenly by this factor, (2-3i) is one of its roots.

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