HELP! Write Polynomial in completely factored form

Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.
Kase082
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HELP! Write Polynomial in completely factored form

Hello,

Any assistance on learning the process for this problem would be greatly appreciated.

For each of the following complex polynomials, one of its zeroes is given. Use this zero to help write the polynomial in completely factored form.
a. C(z) = z^3+(1-4i)z^2+(-6-4i)z+24i ; z=4i

Gotta love having the answer, but no clue how to solve. Can anyone help? THANKS!

FWT
Posts: 153
Joined: Sat Feb 28, 2009 8:53 pm

Re: HELP! Write Polynomial in completely factored form

For each of the following complex polynomials, one of its zeroes is given. Use this zero to help write the polynomial in completely factored form.
a. C(z) = z^3+(1-4i)z^2+(-6-4i)z+24i ; z=4i

Gotta love having the answer, but no clue how to solve.
They give you the one zero (in this degree-3 poly) because then you can divide that out and then use the Quadratic Formula to do the rest. So do the long poly division to take out that one zero / factor (b/c if x = a is a zero then x - a is a factor). Set it up the regular way:

Code: Select all

```long div: -------------------------------------- z - 4i ) z^3 + (1 - 4i)z^2 + (-6 - 4i)z + 24i```
Then do the division the regular way:

Code: Select all

```long div: z^2 -------------------------------------- z - 4i ) z^3 + (1 - 4i)z^2 + (-6 - 4i)z + 24i z^3 + ( - 4i)z^2 z^2 -------------------------------------- z - 4i ) z^3 + (1 - 4i)z^2 + (-6 - 4i)z + 24i z^3 + ( - 4i)z^2 ------------------ (1 )z^2 + (-6 - 4i)z + 24i z^2 + 1z -------------------------------------- z - 4i ) z^3 + (1 - 4i)z^2 + (-6 - 4i)z + 24i z^3 + ( - 4i)z^2 ------------------ (1 )z^2 + (-6 - 4i)z + 24i (1 )z^2 + ( - 4i)z z^2 + 1z -------------------------------------- z - 4i ) z^3 + (1 - 4i)z^2 + (-6 - 4i)z + 24i z^3 + ( - 4i)z^2 ------------------ (1 )z^2 + (-6 - 4i)z + 24i (1 )z^2 + ( - 4i)z ------------------------- (-6 )z + 24i z^2 + 1z + -6 -------------------------------------- z - 4i ) z^3 + (1 - 4i)z^2 + (-6 - 4i)z + 24i z^3 + ( - 4i)z^2 ------------------ (1 )z^2 + (-6 - 4i)z + 24i (1 )z^2 + ( - 4i)z ------------------------- (-6 )z + 24i (-6 )z + 24i```
This leaves you with something you can factor or use the Formula on. So it's what they show here but with complexes.