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can you check my work? i think i got every case though..

ok. YW!

it only works where the angles are all opposite of each other 

i mean x^3-6x^2+3x+10

and synthetic division is just a quicker way of doing long division, its almost the same thing

i dont think it was that bad, and i just learned this :P

the figure is cyclic, so those three angles are just half of the total sum of all the angles, which is 720/2. subtracting, you get 360-105-110=145.

HOPE THIS HELPED!

Use synthetic division twice on the two polynomials to get the whole equation as 18/(x+1).

the roots of the denominator of the original cubic are -1, 2, and 5 so those will not work. the remaining ones are the the positive and negative factors of 18, subtracted by 1 so you get -19, -10, -5, -3, -2, -1(extraneous), 0, 1, 2(extraneous), 5(extraneous), 8, and 17. adding up the ones that work, you get 

-19 - 10 - 5 - 3 - 2 + 0 + 1 + 8 + 17. with a little mental math, you get -13.

HOPE THIS HELPED!

ok just like the other one i did, the angle is 45 degrees. ignoring the diagram(again lol)

z=a+bi

w=c+di

a^2+b^2=16

c^2+d^2=4

if z=-4i then w is -2sqrt2-2isqrt2

divide w/z and take the reciprocal.

w/z = i(-2sqrt2-2isqrt2)/4 or (2sqrt2-2isqrt2)/4 which reduces to (sqrt2-isqrt2)/2.

to get z/w you have to take the reciprocal which is 2/(sqrt2-isqrt2). multiply by conjugate on top and bottom you get 2(sqrt2+isqrt2)/(2+2), which is

(sqrt2+isqrt2)/2 so the imaginary part is sqrt2/2.

HOPE THIS HELPED!!

im assuming it means pi/3 radians, which is 60 degrees.

ok i suck at these, but im gonna give it a try

if z=a+bi, a^2+b^2=9

if w=c+di, c^2+d^2=4

ignoring the diagram, if w is 2i, then z is -3sqrt3/2+3i/2 by 30-60-90. the quotient is i(-3sqrt3/2+3i/2)/(-2), or (-3/2)/(-2) ignoring the complex part, so the answer is 3/4.

HOPE THIS HELPED!!!!

since ad/db is 1/5, then the ratio of the heights of the two triangles if you turned it so that line BEC is horizontal would be 5/(1+5) or 5/6, and the ratio of the bases is 2/(2+3) or 2/5. then, the ratio of the areas DEB/ABC = 5/6*2/5 or 1/3, so the area of triangle ABC is just 48*3 or 144.