Roots Given that 4x316x221x27 has the solution 3 find the co

Roots.

Given that 4x^3-16x^2+21x-27 has the solution 3 find the conjugate complex pair of roots.

The equation 4x^3-16x^2+21x-27=0 has a real root equal to 3.

We can write the equation as:

4x^3 - 16x^2 + 21x - 27 = 0

=> 4x^3 - 12x^2 - 4x^2 + 12x + 9x - 27 = 0

=> 4x^2(x - 3) - 4x(x - 3) + 9(x - 3) = 0

=> (4x^2 - 4x + 9)(x - 3) = 0

The roots of 4x^2 - 4x + 9 = 0 are

x1 = 4/8 + sqrt(16 - 144)/8

=> 1/2 + i*sqrt 2

x2 = 1/2 - i*sqrt 2

The complex roots of the equation 4x^3-16x^2+21x-27=0 are 1/2 + i*sqrt 2 and 1/2 - i*sqrt 2