Question

Which answer best describes the complex zeros of the polynomial function? f(x)=x3−x2+6x−6?

Answer 1

`f(x)=x^3-x^2+6x-6=x^2(x-1)+6(x-1)=(x^2+6)(x-1)=\\\\=\big(x^2-(-6)\big)(x-1)= \big(x^2-(-1\cdot6)\big)(x-1)=\\\\=\big(x^2-i^2(√(6))^2\big)(x-1)= \big(x^2-(i√(6))^2\big)(x-1)=\\\\=\boxed{(x-i√(6))(x+i√(6))(x-1)}`

So complex zeros:


`x_1=-i√(6)`

and


`x_2=i√(6)`

Answer 2

As options are not given, you can match your answer with the option.

The equation of the polynomial is ,
`f(x)=x^(3)-x^(2)+6 x -6`

= x²(x-1)+6(x-1)

= (x² +6)(x-1)

= (x+ i√6)(x-i√6)(x-1)→a²+b²=(a+ib)(a-ib), i²=-1

The two complex zeroes are , put x+i√6=0 ∧ x-i√6=0 is -i√6 and i√6.