To apply the Fundamental Theorem of Algebra, we first need to rewrite the equation in standard polynomial form. The given equation is:
\[ 12x - 6x^2 + 3x^4 = 6x^3 + 2x - x^4. \]
We can rearrange this equation by moving all terms to one side:
\[ 3x^4 + x^4 - 6x^2 - 6x^3 + 12x - 2x = 0. \]
Combining like terms, we get:
\[ 4x^4 - 6x^3 - 6x^2 + 10x = 0. \]
Now, we can factor out a common factor from all terms:
\[ 2x(2x^3 - 3x^2 - 3x + 5) = 0. \]
This gives us one root at \( x = 0 \).
Next, we need to find the number of roots of the cubic polynomial \( 2x^3 - 3x^2 - 3x + 5 \). A cubic polynomial (degree 3) can have:
- 3 real roots,
- 1 real root and 2 complex conjugate roots.
Thus, it will have a total of 3 roots (considering multiplicities).
Adding the root we found, we have:
- 1 root from \( 2x = 0 \),
- 3 roots from the cubic equation.
So, the total number of roots for the original equation is:
\[ 1 + 3 = 4. \]
Therefore, the total number of roots for the equation is 4.
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