Question

Apply the Fundamental Theorem of Algebra to find the number of roots for the equation 12x−6x^2+3x^4=6x^3+2x−x4 .

Answer 1

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

The original equation is:

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

We can first move all terms to one side of the equation:

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

Now, we can combine like terms:

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

This simplifies to:

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

Next, we can factor out a common term from the polynomial:

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

This gives us one root \( x = 0 \). To find the remaining roots, we need to analyze the cubic polynomial \( 2x^3 - 3x^2 - 3x + 5 \).

According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has exactly \( n \) roots (counting multiplicities) in the complex number system. The degree of our original polynomial is 4 (the highest power of \( x \) is 4).

Thus, the total number of roots for the equation \( 4x^4 - 6x^3 - 6x^2 + 10x = 0 \) is:

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

This includes the root \( x = 0 \) and three other roots stemming from the cubic polynomial \( 2x^3 - 3x^2 - 3x + 5 \), which may be real or complex.

In summary, the number of roots for the equation is \( \boxed{4} \).