Im z + Re z (complex numbers) Solution over C of polynomial equations?

Question

Find are 3 factors over C, of -: 2z^3 + 9z^2 + 14z + 5

The problem is, finding the first factor using P(x) [factor theorem], i tried (-1 to -10) and (1 - 10) but none of them have P(x) = 0

sorry, i knows its a dumb question, but all i need to know is how they got the first factor, and then i can get the rest my myself.

And plz explain your answer in step by step, because i know the answer but i don't know how they get it!

plz help, its from my specialist math book (Heinemann VCE zone unit 3&4)

Answer 1

Let p(z) = 2z^3 + 9z^2 + 14z + 5.

What you want to try are all the factors of 5 (your term without a z), divided by all the factors of the coefficient of z^3 (which is 2). You want to try 1, -1, 5, -5, -1/2, 1/2, -5/2, 5/2

Since you said you tried -1 to -10, and 1 to 10, let's reject 1 -1, 5, -5.

Let's try p(-1/2) and see if that gives us 0.

p(-1/2) = 2[-1/2]^3 + 9[-1/2]^2 + 14[-1/2] + 5
p(-1/2) = 2[-1/8] + 9[1/4] - 7 + 5
p(-1/2) = 2[-1/8] + [9/4] - 2
p(-1/2) = [-2/8] + [18/8] - [16/8] = 16/8 - 16/8 = 0

Therefore, since -1/2 is a root, (z + 1/2) is a factor. But you may also make (2z + 1) a factor.

Now that you have one root, you said you can get the others, but I'll work on it anyway so you can compare answers. Read no more if you wish to do this yourself.

You would do long division on (2z^3 + 9z^2 + 14z + 5) / (2z + 1)

Without working out the details of the long division, you should get the answer z^2 + 4z + 5

So your factorization so far would be (2z + 1) (z^2 + 4z + 5).

Equating the second factor to zero, we get

z^2 + 4z + 5 = 0

Instead of using that bulky quadratic formula, I'm going to complete the square.

z^2 + 4z + 4 + 1 = 0
z^2 + 4z + 4 = -1
(z + 2)^2 = -1
z + 2 = +/- i
z = 2 +/- i

So z = {2 + i, 2 - i}, and your factors are:

(2z + 1) (z - (2 + i)) (z - (2 - i)), OR

[2z + 1] [z - 2 - i] [z - 2 + i]

Rule of thumb:

Whenever you have a cubic in the form
ax^3 + bx^2 + cx + d

What you want to do is find ALL the factors of a (let's call them a1, a2), and ALL the factors of d, (let's call them d1, d2, d3) and the rational numbers you want to test would be:

+/- d1/a1, +/- d1/a2, +/- d2/a1, +/- d2/a2, +/- d3/a1, +/- d3/a2

"+/-" means "plus or minus".