Mister Exam

Factor polynomial x^2-x+1

An expression to simplify:

The solution

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 2        
x  - x + 1
(x2x)+1\left(x^{2} - x\right) + 1
x^2 - x + 1
The perfect square
Let's highlight the perfect square of the square three-member
(x2x)+1\left(x^{2} - x\right) + 1
To do this, let's use the formula
ax2+bx+c=a(m+x)2+na x^{2} + b x + c = a \left(m + x\right)^{2} + n
where
m=b2am = \frac{b}{2 a}
n=4acb24an = \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,
(x12)2+34\left(x - \frac{1}{2}\right)^{2} + \frac{3}{4}
Factorization [src]
/              ___\ /              ___\
|      1   I*\/ 3 | |      1   I*\/ 3 |
|x + - - + -------|*|x + - - - -------|
\      2      2   / \      2      2   /
(x+(123i2))(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 + x  - x
x2x+1x^{2} - x + 1
1 + x^2 - x
Powers [src]
     2    
1 + x  - x
x2x+1x^{2} - x + 1
1 + x^2 - x
Combining rational expressions [src]
1 + x*(-1 + x)
x(x1)+1x \left(x - 1\right) + 1
1 + x*(-1 + x)
Rational denominator [src]
     2    
1 + x  - x
x2x+1x^{2} - x + 1
1 + x^2 - x
Trigonometric part [src]
     2    
1 + x  - x
x2x+1x^{2} - x + 1
1 + x^2 - x
Assemble expression [src]
     2    
1 + x  - x
x2x+1x^{2} - x + 1
1 + x^2 - x
Common denominator [src]
     2    
1 + x  - x
x2x+1x^{2} - x + 1
1 + x^2 - x
Combinatorics [src]
     2    
1 + x  - x
x2x+1x^{2} - x + 1
1 + x^2 - x
Numerical answer [src]
1.0 + x^2 - x
1.0 + x^2 - x