The Solution:
The given polynomial is
[tex]P(x)=2x^4-4x^3-16x^2[/tex]A root of the polynomial P(x) is the value of x for which the polynomial P(x) is equal to zero.
That is, any value of x that makes P(x) = 0, is a root of P(x).
The Multiplicity of a Root: This is the number of times a particular root appears as a root in a polynomial.
To find the root of a polynomial, say P(x), you have to equate P(x) to zero, and then solve for the value of x.
So, we shall follow the above procedures to find the root(s) of P(x), and thereafter determine if there are multiple roots.
[tex]\begin{gathered} P(x)=2x^4-4x^3-16x^2=0 \\ \text{Factoring out 2x}^2,\text{ we have} \\ 2x^2(x^2-2x-8)=0 \end{gathered}[/tex]This means that:
[tex]\begin{gathered} x^2-2x-8=0 \\ or \\ 2x^2=0 \end{gathered}[/tex]Solving quadratic equations above by Tthe Factorization Method, we get
[tex]\begin{gathered} x^2-2x-8=0 \\ x^2-4x+2x-8=0 \\ x(x-4)+2(x-4)=0 \\ (x-4)(x+2)=0 \end{gathered}[/tex]So,
[tex]\begin{gathered} P(x)=2x^2(x-4)(x+2)=0 \\ \text{This means} \\ 2x^2=0\text{ }\Rightarrow x=0 \\ x-4=0\text{ }\Rightarrow x=4 \\ x+2=0\text{ }\Rightarrow x=-2 \\ So,\text{ the roots of P(x) are 0, -2, and 4} \end{gathered}[/tex]Looking at the roots of P(x) above, there is no root that appears more than once, hence, the multiplicity of each of the roots is one.