To apply the Fundamental Theorem of Algebra, we first need to rearrange the equation given:
Starting with the equation: \[ 12x - 6x^2 + 3x^4 = 6x^3 + 2x - x^4 \]
We can move all terms to one side of the equation and simplify: \[ 3x^4 + x^4 - 6x^3 - 6x^2 + 12x - 2x = 0 \] \[ 4x^4 - 6x^3 - 6x^2 + 10x = 0 \]
Next, factor out the common term on the left side: \[ 2x(2x^3 - 3x^2 - 3x + 5) = 0 \]
Setting \(2x = 0\) gives us one root: \[ x = 0 \]
Now we need to find the number of roots for the cubic polynomial \(2x^3 - 3x^2 - 3x + 5\). A cubic polynomial can have up to 3 roots (real or complex).
Thus, counting the roots, we have:
- One root from \(x = 0\).
- Up to 3 roots from the cubic polynomial.
According to the Fundamental Theorem of Algebra, the total number of roots (including complex roots) for a polynomial of degree \(n\) is \(n\). Since our polynomial \(2x^3 - 3x^2 - 3x + 5\) is of degree 3, it will have 3 roots (counting multiplicities).
Therefore, the total number of roots for the original equation is: \[ 1 + 3 = 4 \]
In conclusion, the equation has a total of 4 roots.