State the complex and real zeros of this function: X^3-3x^2+x+1

Question

X^3-3x^2+x+1

Answer 1

Given polynomial x^3-3x^2+x+1

= x^3-x^2-2x^2+2x-x+1

= x^2(x-1)-2x(x-1)-1(x-1)

= (x-1)(x^2-2x-1)

so x-1 = 0

x = 1 is one of the real zeros of given polynomial.

Now factorise (x^2-2x-1) = 0

compare quadratic equation ax^2+bx+c = 0

a = 1, b = -2,c = -1

Roots are x = [-b±√(b^2-4ac)]/2a

x = [2±√(4+4)]2

x = [2±√8]/2

x = [2±√4*2]/2

x = [2±2√2]/2

x = 2[1±1.414]/2

x = 1+1.414 and x = 1-1.414

x = 2.414   and x = -0.414

Real zeros of given polynomial function are x = 1, 2.414,-0.414.