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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
General simplification [src] 
         2
1 + z + z
z2+z+1z^{2} + z + 1 
1 + z + z^2
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}
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)
Powers [src] 
         2
1 + z + z
z2+z+1z^{2} + z + 1 
1 + z + z^2
Rational denominator [src] 
         2
1 + z + z
z2+z+1z^{2} + z + 1 
1 + z + z^2
Trigonometric part [src] 
         2
1 + z + z
z2+z+1z^{2} + z + 1 
1 + z + z^2
Common denominator [src] 
         2
1 + z + z
z2+z+1z^{2} + z + 1 
1 + z + z^2
Combinatorics [src] 
         2
1 + z + z
z2+z+1z^{2} + z + 1 
1 + z + z^2
Assemble expression [src] 
         2
1 + z + z
z2+z+1z^{2} + z + 1 
1 + z + z^2
Numerical answer [src] 
1.0 + z + z^2
1.0 + z + z^2
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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