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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
(z2−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   /
(x+(−12−3i2))(x+(−12+3i2))\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
z2−z+1z^{2} - z + 1 
1 + z^2 - z
The perfect square
Let's highlight the perfect square of the square three-member
(z2−z)+1\left(z^{2} - z\right) + 1
To do this, let's use the formula
az2+bz+c=a(m+z)2+na z^{2} + b z + c = a \left(m + z\right)^{2} + n
where
m=b2am = \frac{b}{2 a}
n=4ac−b24an = \frac{4 a c - b^{2}}{4 a}
In this case
a=1a = 1
b=−1b = -1
c=1c = 1
Then
m=−12m = - \frac{1}{2}
n=34n = \frac{3}{4}
So,
(z−12)2+34\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
z2−z+1z^{2} - z + 1 
1 + z^2 - z
Assemble expression [src] 
     2    
1 + z  - z
z2−z+1z^{2} - z + 1 
1 + z^2 - z
Combinatorics [src] 
     2    
1 + z  - z
z2−z+1z^{2} - z + 1 
1 + z^2 - z
Trigonometric part [src] 
     2    
1 + z  - z
z2−z+1z^{2} - z + 1 
1 + z^2 - z
Rational denominator [src] 
     2    
1 + z  - z
z2−z+1z^{2} - z + 1 
1 + z^2 - z
Powers [src] 
     2    
1 + z  - z
z2−z+1z^{2} - z + 1 
1 + z^2 - z
Combining rational expressions [src] 
1 + z*(-1 + z)
z(z−1)+1z \left(z - 1\right) + 1 
1 + z*(-1 + z)
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