Find a polynomial with real coefficients that has the given

Find a polynomial with real coefficients that has the given zeros. -1 and 5 - 2i One such polynomial P(x) can be defined as P(x) = x^3 - 9x^2 + x + 29.

Solution

Let P(x) be the polynomial whose two zeros are - 1 and 5 - 2i then the third zero is nothing but the conjugate of 5 - 2i that is 5 + 2i.

So the polynomial P(x) is

P(x) = ( x + 1 )( x - 5 + 2i )( x - 5 - 2i)

        = ( x + 1 )(( x - 5 )² - ( 2i )² )

        =( x + 1 )(x² + 25 - 10x + 4 )     since i² = - 1

        = ( x + 1 )( x² - 10x + 29 )

        = x³ - 10x² + 29x + x² - 10x + 29

        = x³ - 9x² + 19x + 29

Hence our required polynomial is P(x) = x³ - 9x² + 19x + 29