To apply the Fundamental Theorem of Algebra to the equation
\[ 12x - 6x^2 + 3x^4 = 6x^3 + 2x - x^4, \]
we first need to rearrange the equation into standard polynomial form by moving 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 greatest common factor, which is 2:
\[ 2(2x^4 - 3x^3 - 3x^2 + 5x) = 0. \]
From here, we can determine the number of roots by analyzing the polynomial \( 2x^4 - 3x^3 - 3x^2 + 5x \).
The degree of the polynomial \( 2x^4 - 3x^3 - 3x^2 + 5x \) is 4. According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has exactly \( n \) roots in the complex number system (counting multiplicities).
Therefore, in this case, since the polynomial is of degree 4, we can conclude that it has:
4 roots.
Thus, the answer is:
E. 4
1 answer