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

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

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)