What is the fundamental theorem of algebra of this polynomial in completed factored form? and how do I graph it?

f(x)=x^5-x^4+x^3-x^2+x-1

it says there should be 5 roots.

f(1) = 1-1+1-1+1-1 = 0
so (x-1) is a factor
by synthetic division I had
(x-1)(x^4 + x^2 + 1)

x^2 = (-1 ± √-3)/2 = (-1 ± i√3)/2

now take √ of that again.
So we get one real root of x=1 and 4 imaginary roots.

The fundamental theorem of algebra states that every polynomial equation of degree n has n complex roots, taking into account multiplicity. In the case of the polynomial you provided, f(x) = x^5 - x^4 + x^3 - x^2 + x - 1, it is a polynomial equation of degree 5. Therefore, it has 5 complex roots, considering multiplicity.

To graph the polynomial, follow these steps:

1. Determine the x-intercepts: Set f(x) = 0 and solve for x. The x-intercepts are the values of x where the graph of the polynomial crosses the x-axis.

2. Find the y-intercept: Substitute x = 0 into the equation f(x) = x^5 - x^4 + x^3 - x^2 + x - 1 to determine the point where the graph crosses the y-axis.

3. Determine the behavior at the extremes: As x approaches positive or negative infinity, observe the end behavior of the polynomial. In this case, since the degree is odd (5), the graph will approach positive infinity as x goes to positive infinity and negative infinity as x goes to negative infinity.

4. Locate additional points: You can locate additional points by substituting different x-values into the equation and calculating the corresponding y-values. You may choose some x-values symmetric around the y-axis to simplify the process.

5. Plot the points: Use the x and y-values obtained from the previous steps to plot the points on a graph.

6. Draw the curve: Connect the plotted points smoothly to sketch the curve of the polynomial.

Please note that for higher-degree polynomials, it can be challenging to accurately graph them by hand without using technology or calculators.

The fundamental theorem of algebra states that every polynomial equation of degree n has exactly n complex solutions (counting multiplicity).

To find the fundamental theorem of algebra for the given polynomial f(x) = x^5 - x^4 + x^3 - x^2 + x - 1 in completed factored form, we need to determine the roots of the polynomial.

To graph the polynomial, we will follow these steps:

1. Find the x-intercepts (or roots) of the polynomial by setting f(x) equal to zero and solving for x. In this case, we need to find the values of x such that: x^5 - x^4 + x^3 - x^2 + x - 1 = 0.

2. Use factoring, synthetic division, or numerical methods such as the Newton-Raphson method to find the roots. In this case, the polynomial is not easily factored and synthetic division might be time-consuming, so we can use numerical methods or a graphing calculator to find the approximate values of the roots.

3. Once you have found the roots, plot them on a coordinate plane. These will be the x-intercepts of the graph.

4. Determine the behavior of the polynomial for large positive and negative values of x. By observing the powers of x in the polynomial, we can conclude that as x approaches negative infinity, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches positive infinity.

5. Plot a few additional points to get a sense of the shape of the graph. Choose values of x and calculate the corresponding values of f(x) to plot.

6. Connect the plotted points smoothly to form the graph of the polynomial.

Please note that graphing a polynomial manually can be a time-consuming task, so it is often more efficient to use graphing software or online graphing tools to graph the polynomial accurately and quickly.