Question

Apply the Fundamental Theorem of Algebra to find the number of roots for the equation 12x−6x2+3x4=6x3+2x−x4 .(1 point)

Answer 1

To apply the Fundamental Theorem of Algebra, we first need to rearrange the equation given:

Starting with the equation:
\[
12x - 6x^2 + 3x^4 = 6x^3 + 2x - x^4
\]

We can move all terms to one side of the equation and simplify:
\[
3x^4 + x^4 - 6x^3 - 6x^2 + 12x - 2x = 0
\]
\[
4x^4 - 6x^3 - 6x^2 + 10x = 0
\]

Next, factor out the common term on the left side:
\[
2x(2x^3 - 3x^2 - 3x + 5) = 0
\]

Setting \(2x = 0\) gives us one root:
\[
x = 0
\]

Now we need to find the number of roots for the cubic polynomial \(2x^3 - 3x^2 - 3x + 5\). A cubic polynomial can have up to 3 roots (real or complex).

Thus, counting the roots, we have:
1. One root from \(x = 0\).
2. Up to 3 roots from the cubic polynomial.

According to the Fundamental Theorem of Algebra, the total number of roots (including complex roots) for a polynomial of degree \(n\) is \(n\). Since our polynomial \(2x^3 - 3x^2 - 3x + 5\) is of degree 3, it will have 3 roots (counting multiplicities).

Therefore, the total number of roots for the original equation is:
\[
1 + 3 = 4
\]

In conclusion, the equation has a total of **4 roots**.