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

The fourth root is 3

If the 4th root is not a real therefore it must be a  complex number (a+ib),and its conjugate will be also a root ,therefore there would be 5 roots instead of 4  roots.

Therefore the fourth root is real.

The roots are -1 with multiplicity 2 and 3 with multiplicity 2

Therefore it has four roots

Step-by-step explanation:

Given polynomial equation is

And also given that 3,-1 and -1 are  the roots of the given polynomial equation

To find the fourth root of the polynomial equation and to solve the fourth root is real :

By synthetic division

_3|   1    -4    -2    12    9

       0    3    -3   -15    -9

___________________

_-1|       1     -1     -5    -3     0

            0     -1      2     3

___________________

            1      -2     -3     0

Therefore x-3 and x+1 is a factor

Therefore 3 and -1 are roots

Now we have the quadratic equation

Therefore x=-1,3 are the roots

Therefore the fourth root is 3

If the 4th root is not a real therefore it must be a  complex number (a+ib),and its conjugate will be also a root ,therefore there would be 5 roots instead of 4  roots.

Therefore the fourth root is real.

The roots are -1 with multiplicity 2 and 3 with multiplicity 2

Therefore it has four roots.

a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. because no real number satisfies this equation, i is called an imaginary number. has no real solution, since the square of a real number cannot be negative.

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