Question

Three roots of the polynomial equation x^4 - 4x^3 - 2x^2 + 12x + 9 = 0 are 3, -1, and -1. Explain why the forth root must be a real number. Find the forth root.

Answer 1

Hello,

This polynomial function has maximum 4 roots as the degree is 4

and their coefficients are all reals so if one root is a complex number, its conjugate is a root too.

As we know already 3 roots and none of them is a complex number, the fourth one is a real number

And their product is 9 = 3 * 1 * (-1) * (-3) as

`(x-x_1)(x-x_2)(x-x_3)(x-x_4)=x^4+...+x_1x_2x_3x_4`

(The constant term is 9, the product of the roots)

so the last root is -3

Thanks