Question

apply the fundamental theorem of algebra to find the number of roots for the equation $12x - 6x^2 + 3x^4 = 6x^3 + 2x - x^4$. (1 point)
\boxed{} roots

Explanation:

Step1: Rearrange the equation

First, we need to move all terms to one side of the equation to get a polynomial in standard form.
Starting with \(12x - 6x^{2}+3x^{4}=6x^{3}+2x - x^{4}\), we add \(x^{4}\) to both sides, subtract \(6x^{3}\) from both sides, subtract \(2x\) from both sides:

$$
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Step2: Identify the degree of the polynomial

The degree of a polynomial is the highest power of \(x\) with a non - zero coefficient. In the polynomial \(4x^{4}-6x^{3}-6x^{2}+10x = 0\), the highest power of \(x\) is \(4\) (from the term \(4x^{4}\)).

Step3: Apply the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) (where \(n>0\)) has exactly \(n\) roots (counting multiplicities) in the complex number system. Since our polynomial is of degree \(4\), it has \(4\) roots.

Answer:

\(4\)