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, let's rewrite the equation as:
3x^4 + x^4 + 6x^3 + 12x - 6x^2 - 2x = 0
We can simplify this equation by combining like terms:
4x^4 + 6x^3 - 6x^2 + 10x = 0
Now, let's count the total degree of the polynomial, which is the highest power of x:
Total degree = 4
According to the fundamental theorem of algebra, the polynomial equation of degree n has exactly n roots, counted with multiplicity.
Therefore, this equation has 4 roots.