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 the discriminant to determine if the quadratic can be factored over the real numbers. The discriminant is calculated as D=b2−4*a*c

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

4. Conclude that since the discriminant is negative (D<0, the quadratic has no real roots and cannot be factored into linear factors with real coefficients. It is considered an irreducible quadratic over the real numbers.

5. Factor over the complex numbers if required using the quadratic formula x=(−b±√(,D))/(2*a) The roots are x=(−1±i√(,3))/2

6. Write the factored form using the complex 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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