Answer:
[tex] 3 {x}^{3} - 25 {x}^{2} + 35x - 23 = 0[/tex]
Step-by-step explanation:
We want to create an equation that will have the solution
[tex]x = \frac{1}{3},x = 4 + \sqrt{7}i,x = 4 - \sqrt{7}i[/tex]
This implies that:
[tex]3x -1= 0,x -( 4 + \sqrt{7}i) = 0,x - ( 4 - \sqrt{7}i) = 0[/tex]
We put the roots in factored form by reversing the zero product principle to get:
[tex](3x -1)(x -( 4 + \sqrt{7}i))(x - ( 4 - \sqrt{7}i)) = 0[/tex]
We expand the last two parenthesis to get:
[tex](3x - 1)( {x}^{2} - 4x + \sqrt{7}ix - 4x - \sqrt{7}ix + 16 - 7 {i}^{2}) = 0 [/tex]
We simplify to get:
[tex](3x - 1)( {x}^{2} - 8x + 23) = 0[/tex]
We expand further to obtain:
[tex]3 {x}^{3} - 25 {x}^{2} +35x - 23 = 0[/tex]
