the sign of a real function

Question

Let us define:

$$P(m)=96 m^8 - 2112 m^7 + 19392m^6-79872 m^5 + 117600 m^4 +17472 m^3 -162432 m^2+99072 m-9216$$

for

$$m \in \mathbb{R}$$

How can I show that: $P(m)>0 $ for $ m \geq 9 $.

PS: Please note that I have already drawn the graph of $P$ with "MAPLE" and it gives what I want to proof.

Answer 1

If you show it is positive for a value $m>9$ and then show that all $9$ roots are $\le 9$ then the values cannot proceed below the vertical axis.

Answer 2

As a polynomial of degree $8$ with a positive leading coefficient, we know that it opens up towards the positive y-axis after its last root is achieved.

Note that

$$P(m) = 96 (m - 1)^2 (m - 2) (m^5 - 18 m^4 + 125 m^3 - 240 m^2 - 396 m + 48)$$

The irreducible polynomial of degree $5$ has roots that can be numerically confirmed to have the following approximate values:

$$x=-1.09834$$ $$x=0.113819$$ $$x=5.53445$$ $$x = 6.72504 + 4.91435 i$$ $$x = 6.72504 - 4.91435 i$$

And the largest of these roots is less than $9$, hence $P(m) > 0$ for $x \geq 9$.

(In fact, $P(m) > 0$ for $x > 5.54$)

Answer 3

i have use a numerical method, here are all roots in an approximative form $$-2.15533752664163, 0.0277005296499047-0.0662778779440053i, 0.0277005296499047+0.0662778779440053i, 1.21497451131535-3.57762045136675i, 1.21497451131535+3.57762045136675i, 6.94587700876063-5.43061456034008i, 6.94587700876063+5.43061456034008i, 7.77823342718986$$