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Expression simplification/ Factorization polynomials/ z^2-z+1

Factor polynomial z^2-z+1

An expression to simplify:

The solution

You have entered [src] 
 2        
z  - z + 1
$$\left(z^{2} - z\right) + 1$$ 
z^2 - z + 1
Factorization [src] 
/              ___\ /              ___\
|      1   I*\/ 3 | |      1   I*\/ 3 |
|x + - - + -------|*|x + - - - -------|
\      2      2   / \      2      2   /
$$\left(x + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)\right) \left(x + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)\right)$$ 
(x - 1/2 + i*sqrt(3)/2)*(x - 1/2 - i*sqrt(3)/2)
General simplification [src] 
     2    
1 + z  - z
$$z^{2} - z + 1$$ 
1 + z^2 - z
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(z^{2} - z\right) + 1$$
To do this, let's use the formula
$$a z^{2} + b z + c = a \left(m + z\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = -1$$
$$c = 1$$
Then
$$m = - \frac{1}{2}$$
$$n = \frac{3}{4}$$
So,
$$\left(z - \frac{1}{2}\right)^{2} + \frac{3}{4}$$
Numerical answer [src] 
1.0 + z^2 - z
1.0 + z^2 - z
Common denominator [src] 
     2    
1 + z  - z
$$z^{2} - z + 1$$ 
1 + z^2 - z
Assemble expression [src] 
     2    
1 + z  - z
$$z^{2} - z + 1$$ 
1 + z^2 - z
Combinatorics [src] 
     2    
1 + z  - z
$$z^{2} - z + 1$$ 
1 + z^2 - z
Trigonometric part [src] 
     2    
1 + z  - z
$$z^{2} - z + 1$$ 
1 + z^2 - z
Rational denominator [src] 
     2    
1 + z  - z
$$z^{2} - z + 1$$ 
1 + z^2 - z
Powers [src] 
     2    
1 + z  - z
$$z^{2} - z + 1$$ 
1 + z^2 - z
Combining rational expressions [src] 
1 + z*(-1 + z)
$$z \left(z - 1\right) + 1$$ 
1 + z*(-1 + z)
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