If x^15-x^13+x^11-x^9+x7-x^5+x^3-x = 7 prove x^16 > 15

Given , we are asked to show that :

First, note that so the polynomial has real roots at -1,0, and 1.

For x<-1 the polynomial is negative.

For -1<x<0 the polynomial is positive but has a maximum a little less than 1/2.

For 0<x<1 the polynomial is negative.

So we know that x>1 for the polynomial to achieve 7.

Since x>1 we have

Given , we are asked to show that :

First, note that so the polynomial has real roots at -1,0, and 1.

For x<-1 the polynomial is negative.

For -1<x<0 the polynomial is positive but has a maximum a little less than 1/2.

For 0<x<1 the polynomial is negative.

So we know that x>1 for the polynomial to achieve 7.

Since x>1 we have

Now multiply both sides of the equation by :

Multiplying and factoring we get:

Then:

  ; but so

and

as required.

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and

so multiplying the degree 15 polynomial by x^2+1 gives x times the degree 16 polynomial.