Answer:
Part 1) ![(x+4)(x-1)(x-2)(x-4)](https://tex.z-dn.net/?f=%28x%2B4%29%28x-1%29%28x-2%29%28x-4%29)
The related polynomial equation has a total of four roots, all four roots are real
Part 2) ![(x+1)(x-1)(x+2)^{2}](https://tex.z-dn.net/?f=%28x%2B1%29%28x-1%29%28x%2B2%29%5E%7B2%7D)
The related polynomial equation has a total of four roots, all four roots are real and one root has a multiplicity of 2
Part 3) ![(x+3)(x-4)(x-(2-i))(x+(2-i))](https://tex.z-dn.net/?f=%28x%2B3%29%28x-4%29%28x-%282-i%29%29%28x%2B%282-i%29%29)
The related polynomial equation has a total of four roots, two roots are complex and two roots are real
Part 4) ![(x+i)(x-i)(x+2)^{2}](https://tex.z-dn.net/?f=%28x%2Bi%29%28x-i%29%28x%2B2%29%5E%7B2%7D)
The related polynomial equation has a total of four roots, two roots are complex and one root is real with a a multiplicity of 2
Step-by-step explanation:
we know that
The Fundamental Theorem of Algebra states that: Any polynomial of degree n has n roots
so
Part 1) we have
![(x+4)(x-1)(x-2)(x-4)](https://tex.z-dn.net/?f=%28x%2B4%29%28x-1%29%28x-2%29%28x-4%29)
The roots of this polynomial are
x=-4, x=1,x=2,x=4
therefore
The related polynomial equation has a total of four roots, all four roots are real
Part 2) we have
![(x+1)(x-1)(x+2)^{2}](https://tex.z-dn.net/?f=%28x%2B1%29%28x-1%29%28x%2B2%29%5E%7B2%7D)
The roots of this polynomial are
x=-1, x=1,x=-2,x=-2
therefore
The related polynomial equation has a total of four roots, all four roots are real and one root has a multiplicity of 2
Part 3) we have
![(x+3)(x-4)(x-(2-i))(x+(2-i))](https://tex.z-dn.net/?f=%28x%2B3%29%28x-4%29%28x-%282-i%29%29%28x%2B%282-i%29%29)
The roots of this polynomial are
x=-3, x=4,x=(2-i),x=-(2-i)
therefore
The related polynomial equation has a total of four roots, two roots are complex and two roots are real
Part 4) we have
![(x+i)(x-i)(x+2)^{2}](https://tex.z-dn.net/?f=%28x%2Bi%29%28x-i%29%28x%2B2%29%5E%7B2%7D)
The roots of this polynomial are
x=-i, x=i,x=-2,x=-2
therefore
The related polynomial equation has a total of four roots, two roots are complex and one root is real with a a multiplicity of 2