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.