list all roots of

1) x^5-3x^4+3x^3-x^2=0

2)8x^3-12x^2+6x-1

3) x^4=16

Sorry, I don't remember how to do #1 and 2, but 3 is x=+-2

Finding roots of Eqs that are 3rd degree or higher is done mostly by trial and error.However, Excel Spread sheets made the task a little easier.

First, we find a number that will satisfy the given Eq. It was determined
that 0 and 1 were the only numbers that
satisfied your Eq. so the roots are:

1. x = 0, and x = 1.

2. I found only one root for this one:
X = 0.50.

3. X^4 = 16.
Take sqrt of both sides:
X^2 = 4,
Take sqrt of both sides and get:
x = +- 2.

Sure! Let's solve each equation individually to find all the roots.

1) To find the roots of the equation x^5 - 3x^4 + 3x^3 - x^2 = 0, we can first factor out an x^2, so the equation becomes x^2(x^3 - 3x^2 + 3x - 1) = 0.

Next, we can solve the quadratic equation x^2 = 0, which has a double root at x = 0.

Now, let's solve the cubic equation x^3 - 3x^2 + 3x - 1 = 0. We can use synthetic division or other methods to check for rational roots. We can see that x = 1 is a root because when we substitute it into the equation, we get 1 - 3 + 3 - 1 = 0.

Using long division or synthetic division, we can divide the quadratic equation x^3 - 3x^2 + 3x - 1 by (x - 1). This gives us (x - 1)(x^2 - 2x + 1) = 0.

Factoring the quadratic expression x^2 - 2x + 1 = 0 gives us (x - 1)(x - 1) = 0.

Therefore, the roots of the equation x^5 - 3x^4 + 3x^3 - x^2 = 0 are x = 0 (double root) and x = 1 (triple root).

2) To find the roots of the equation 8x^3 - 12x^2 + 6x - 1 = 0, we can use various methods such as factoring, synthetic division, or numerical methods like Newton's method.

Unfortunately, this equation cannot be easily factored, and synthetic division might not yield rational roots. Therefore, we will use numerical methods to approximate the roots.

One approach is to use a graphing calculator or software to plot the equation and find the x-intercepts. Another approach is to use a numerical root-finding algorithm, such as Newton's method, to approximate the roots.

Using numerical methods or software, we find that the roots of the equation 8x^3 - 12x^2 + 6x - 1 = 0 are approximately x ≈ 0.896, x ≈ 0.104, and x ≈ 0.000.

3) To find the roots of the equation x^4 = 16, we need to solve for x.

Taking the square root of both sides of the equation, we get √(x^4) = ±√16. This simplifies to x^2 = ±4.

From here, we can solve for x by taking the square root of both sides again.

For x^2 = 4, we have two solutions: x = √4 = 2 and x = -√4 = -2.

For x^2 = -4, we have two complex solutions: x = √(-4) = 2i and x = -√(-4) = -2i.

Therefore, the roots of the equation x^4 = 16 are x = 2, x = -2, x = 2i, and x = -2i.