Apply the Fundamental Theorem of Algebra to find the number of roots for the equation 12x - 6x^2 + 3x^4 = 6x^3 + 2x -x^4.
To apply the Fundamental Theorem of Algebra, we first rearrange the given equation:
3x^4 - x^4 + 6x^3 - 12x - 6x^2 + 2x = 0
Combining like terms, we have:
2x^4 + 6x^3 - 6x^2 - 10x = 0
Next, we equate this equation to zero to find the number of roots:
2x^4 + 6x^3 - 6x^2 - 10x = 0
Now, we can see that the degree of the polynomial is 4. According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n roots counting multiplicities. Therefore, this equation has exactly 4 roots.