Question

For the following polynomial one zero is listed. List all zeros.

Answer 1

\[P(x) = x^3+x^2-4x-24\]

Answer 2

\[-2+2i\]

Answer 3

^Is a zero

Answer 4

then you know the complex conjugate is a root:
-2-2i is also a root

Answer 5

Yes, but there is another root I don't know. :c

Answer 6

I think it's 3 or -3, but I'm not sure which.

Answer 7

maybe the easiest way to find the third root is first multiply out the first 2 roots

(x - (-2+2i))(x- (-2-2i))
it is a bit ugly, but if we do it carefully:
x^2 +(-2+2i)(-2-2i) -x(-2+2i)- x(-2-2i)
can you simplify this?
if you do, then divide it into the original expression

Answer 8

if you have a guess, try it in the equation. try 3:
x^3 + x^2 -4x -24

replace x with 3 (this is easy)
3^3 +3^2 -4*3 -24
now do the arithmetic
3*3*3 +3*3 -12 -24
27+9 -36
36-36
0
yes 3 works!

Answer 9

Thanks.

Answer 10

here is the other way
x^2 +(-2+2i)(-2-2i) -x(-2+2i)- x(-2-2i)

(-2+2i)(-2-2i)= (-2)^2 + (-4*i*i)= 4+4=8
-x(-2+2i)= 2x -2xi
- x(-2-2i)= 2x +2xi

add up the parts: x^2+4x+8
divide this into x^3 + x^2 -4x -24