Question 1189226: Please explain how to solve this: https://docs.google.com/document/d/1jpqO35zDSlUDFyoB_pZ0TZF8JsOYXjmlSxNh7r31O-s/edit?usp=sharing
Found 2 solutions by Boreal, ikleyn:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! 
Using synthetic division 3 times with 1 as the root, or polynomial division, either z-1 3 times or z^3-3z^2+3z-1 once, the second term is z^2+3z+6
The roots of that are (1/2)[-3 +/- I sqrt(15)]
All the roots are 1,1,1,-3+ Isqrt(15), and -3-I sqrt(15)
The product of the complex roots is 9 - I^2*15=9+15=24.
The product of the roots is 24.
The sum of the roots is 1+1+1-3+I sqrt (15)-3 - I sqrt (15)=-3
Answer by ikleyn(52814)  (Show Source):
You can put this solution on YOUR website! .
Consider the polynomial P(z) = z^5 - 10z^2 + 15z - 6 of complex variable z.
The polynomial can be written in the form P(z) = (z-1)^3*(z^2 + bz+ c).
Consider the function q(x) = x^5 - 10x^2 + 15x - 6, for real x.
a. Write down the sum and the product of the roots of P(z).
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The answer in the post by @Boreal is INCORRECT.
I came to bring a correct solution with correct answer.
Notice that the problem is too wordy.
Its 1st line combined with the 4th line is totally enough to present a correct formulation.
The 2-nd line and the 3rd line are unnecessary and excessive.
The most interesting fact is that to get the answer, you do not need find the roots
and/or factorize the polynomial in an explicit form.
It is enough to apply the Vieta's theorem. It says that the sum of the roots of the given polynomial equals
to the coefficient at x^4, taken with the opposite sign.
In our case, the coefficient at x^4 is 0 (zero, ZERO); so, the sum of the roots equals 0 (zero, ZERO).
Regarding the product of the roots, apply the Vieta's theorem again. It says that the product of the roots
of the given polynomial equals to the constant term, taken with the opposite sign.
In our case, the constant term is -6; hence, the product of the roots is 6.
ANSWER. The sum of the roots is 0 (zero, ZERO). The product of the roots is 6.
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Solved and thoroughly/carefully/comprehensively explained.
On Vieta's theorem, see this link
https://en.wikipedia.org/wiki/Vieta%27s_formulas
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The solution by @Boreal has arithmetic mistakes, that lead to incorrect answer.
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