Need to solve the inequality below using only algebra ...

-14x^9 + 25x^8 + 414x^7 - 596x^6 - 3434x^5 +2469x^4 + 9414x^3 + 2862x^2 - 1620x <= 0

I can factor out an x to get one root x=0. I graphed it using Quickgraph to get where it crosses the x axis and can see the intervals where it alternates between positive and negative but need to do it algebraically.

They show how to solve polynomials

here. Use the graph to get some easy zeros => easy factors. You already got x=0 so x is a factor (do negative so the rest is easier)

(x)(14x^8 - 25x^7 - 414x^6 + 596x^5 + 3434x^4 - 2469x^3 - 9414x^2 - 2862x + 1620)

My graphing calc "table" shows x=3 is a zero so x - 3 is a factor. Do the long division (or

synthetic division) for this. Also x=-1 so x + 1 is a factor. Do that one too. This should get you down to this:

(x)(x - 3)(x + 1)(14x^6 + 3x^5 - 366x^4 - 127x^3 + 2082x^2 + 1314x - 540)

Then do the big part again in your calculator. I get more zeros at x=-1 and x=3 again. When I take those out I get:

(x)(x - 3)(x + 1)(x - 3)(x + 1)(14x^4 + 31x^3 - 262x^2 - 558x + 180)

The graph of just the big part doesn't show nice solutions, but you can use

the Rational Root Test to look for fraction solutions between -5 & -4, -3 & -2, 0 & 1, and 4 & 5.