what are the roots of x^5-10x^3+9x?

To find the roots of the polynomial x^5 - 10x^3 + 9x, we need to solve the equation x^5 - 10x^3 + 9x = 0.

One way to approach this problem is by factoring out common terms. Notice that each term contains an x, so we can factor out x from each term. This gives us x(x^4 - 10x^2 + 9) = 0.

Now, we have two possibilities for the equation to be true:

1) x = 0
2) x^4 - 10x^2 + 9 = 0

For the first possibility, x = 0 is a root of the polynomial.

Now, let's solve the second equation x^4 - 10x^2 + 9 = 0. This is a quadratic equation in x^2. Let's substitute y = x^2, so the equation becomes y^2 - 10y + 9 = 0.

We can solve this quadratic equation by factoring. The factors of 9 that could add up to -10 are -1 and -9. So, we can rewrite the equation as (y - 1)(y - 9) = 0.

Now, we set each factor to zero and solve for y:

1) y - 1 = 0 => y = 1
2) y - 9 = 0 => y = 9

Since y = x^2, we can substitute back to find the values for x:
1) x^2 = 1 => x = ±√1 => x = ±1
2) x^2 = 9 => x = ±√9 => x = ±3

Therefore, the roots of the polynomial x^5 - 10x^3 + 9x are: x = 0, x = ±1, and x = ±3.