To apply the Fundamental Theorem of Algebra to the equation \( 4x - 6x^3 + 2x^7 = 8x^5 + 3x^6 \), we first need to rearrange it into standard polynomial form.
Starting with: \[ 4x - 6x^3 + 2x^7 = 8x^5 + 3x^6 \] we can move all terms to one side of the equation: \[ 2x^7 - 3x^6 - 8x^5 - 6x^3 + 4x = 0 \]
Next, we write it as: \[ 2x^7 - 3x^6 - 8x^5 - 6x^3 + 4x = 0 \]
Now we have a polynomial \( f(x) = 2x^7 - 3x^6 - 8x^5 - 6x^3 + 4x \). This is a polynomial of degree 7, since the term \( 2x^7 \) has the highest exponent.
According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has exactly \( n \) roots in the complex number system, counting multiplicities.
Thus, for our polynomial \( f(x) \) of degree 7, there are a total of 7 roots, which may be real or complex.