Apply the Fundamental Theorem of Algebra to find the number of roots for the equation 12x−6x2+3x4=6x3+2x−x4

To apply the Fundamental Theorem of Algebra, we first need to rearrange the given equation into a standard polynomial form.

The original equation is:

\[ 12x - 6x^2 + 3x^4 = 6x^3 + 2x - x^4 \]

First, let's move all terms to one side of the equation:

\[ 3x^4 + x^4 - 6x^3 - 6x^2 + 12x - 2x = 0 \]

Combining like terms gives:

\[ 4x^4 - 6x^3 - 6x^2 + 10x = 0 \]

Now, we can factor out the common term \(2x\):

\[ 2x(2x^3 - 3x^2 - 3x + 5) = 0 \]

At this point, we can see that one root is \(x = 0\).

Now, we examine the cubic polynomial \(2x^3 - 3x^2 - 3x + 5\) to find its roots. A cubic polynomial can have either 1 or 3 real roots (counting multiplicities), so we need to check the degree of this polynomial.

Degree Analysis

1. The entire polynomial \(4x^4 - 6x^3 - 6x^2 + 10x\) is of degree 4.
2. According to the Fundamental Theorem of Algebra, a polynomial of degree \(n\) has exactly \(n\) roots, counting multiplicities.

Given that the polynomial is of degree 4, the entire equation can have a total of 4 roots (including multiplicities).

Thus, the number of roots for the original equation is:

\[ \text{Number of roots} = 4 \]