Question

A polynomial with rational coefficients has roots 5 and -6i. (The i is an imaginary number not variable). What is the polynomial? With work shown please

Answer 1

Answer:

The answer is  {$x^{3}$-(5)$x^{2}$ (36)× $x$-(180)

Step-by-step explanation:

   A polynomial with rational coefficients has roots 5 and -6i (There i is the    

   imaginary number not variable )

   As we knew that imaginary no comes in pair . This means that if  -6i is

   one root then the other root will be 6i

   So if we assume the lowest polynomial that is possible is given as

    {$x^{3}$-(sum of all roots)$x^{2}$ +(roots taken two at a time)

    $x$-(product of all roots)

    {$x^{3}$-(5+6i+(-6i))$x^{2}$ +(5×6i +5×(-6i) +6i×(-6i))

    $x$-(5×6i×(-6i))

    $i^2$ = -1

    {$x^{3}$-(5)$x^{2}$ (36)× $x$-(180)

   The general case we assume that the three previous roots and rest roots

   a4,a5,a6 ...............an

  $x^n$-(sum of all roots)×$x^{n-1}$ +(roots taken two at a  

   time) ×$x^{n-2}$- .......................................... (product of all roots)