Factorize the polynomial using complex coefficients

Factorize the polynomials using complex coefficients:
1. z^2+4
2. z^2-iz+2

i have no idea how to do these types of questions. can anyone help?

Hello,

No need for trial and error. Just follow the recipe:

z² + 4
  = z² + 2²
  = z² - (2i)² →→→ Because -1=i²
  = (z + 2i)(z - 2i) →→→ Because a²-b²=(a+b)(a-b)

z² - iz + 2
  = z² - 2×z×(i/2) + (i/2)² + ¼ + 2 →→→ Completing the square
  = (z - i/2)² + 9/4
  = (z - i/2)² + (3/2)²
  = (z - i/2)² - (3i/2)² →→→ Because -1=i²
  = (z - i/2 + 3i/2)(z - i/2 - 3i/2) →→→ Because a²-b²=(a+b)(a-b)
  = (z + 2i/2)(z - 4i/2)
  = (z + i)(z - 2i)

Remember:
● Complete the square;
● Square root the resulting constant;
● Use a²-b² identity to factor.

Regards,
Dragon.Jade :-)

1)....(z + 2i)(z - 2i)

2)....(z - 2i)(z + i)...........THANKS, I LEARNED SOMETHING
......................................… DID A PROBLEM LIKE THIS BEFORE

Mathcerely, Robert Jones "Teacher/Tutor of Fine Students"
Moved Florida to France June 2011 but still tutor world-wide (USA , UK ,
France , Checz Republic ...) free using Skype. If you need me to explain
further or any other math problem just let me know how to get in touch.

********* A PRESENT FOR YOU
Ask someone to write a number, say a five-digit number.
(Can be a 2-digit.....3-digit.....4-digit....ect..…
Suppose the number written is
57836
Now, without showing the asker, you write a number on
a sheet of paper and keep it folded. You have to write the
number by subtracting 2 from the above number and
adding 2 in front which will be....
257834
Next, ask the person to write another five-digit number
below his original number. Suppose he writes 37589.
So you now have...
57836
37589
Now you write a five-digit number below it in such a
way that each digit is 9 minus digit above. You now have...
57836
37589
62410
Ask the person to add one more 5-digit number. If he
adds 54732, you add below it 45267. Note that you decided
the number by subtracting each of his digit from 9.
Thus, you now have.....
57836
37589
62410
54732
45267
Next, ask him to add all the number and he gets 257834
Show him the number which you had written as an answer
to this addition earlier in the folded paper and surprise him.

1. z^2 + 4 = (z - 2i)(z + 2i) by a simple trial and error

2. The roots are z = [i +/- sqrt(-1 - 8)]/2 = [i +/- 3i]/2 = 2i or -i

The factorization then will be z^2 - iz + 2 = (z - 2i)(z + i)