Factor x^2-x+1

Problem

x2−x+1

Solution

1. Identify the type of polynomial. This is a quadratic expression of the form a*x2+b*x+c where a=1 b=−1 and c=1

2. Check for real roots using the discriminant formula D=b2−4*a*c

3. Calculate the discriminant: D=(−1)2−4*(1)*(1)=1−4=−3

4. Determine factorability over real numbers. Since the discriminant is negative (D<0, the quadratic has no real roots and is considered irreducible over the field of real numbers.

5. Apply the quadratic formula x=(−b±√(,D))/(2*a) to find complex roots if factoring over the complex field is required.

6. Substitute the values: x=(−(−1)±√(,−3))/(2*(1))=(1±i√(,3))/2

7. Write the factors using the roots (x_1)=(1+i√(,3))/2 and (x_2)=(1−i√(,3))/2

Final Answer

x2−x+1=(x−(1+i√(,3))/2)*(x−(1−i√(,3))/2)

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