Algebra polynomial

Question

if $P(x) = (2x-5)^{2017}+(2x-5)^{2015}+(x-4)^{2017}+(x-4)^{2015}+3x-9 = 0$ find the real roots. I found one real root $3$ by making sum of polynomial equating to o and finding the value of $x$ but not able to proceed further if any other root exit in real.

Answer 1

It's $$3(x-3)\left((2x-5)^{2016}-(2x-5)^{2015}(x-4)+...+(x-4)^{2016}+(2x-5)^{2014}-(2x-5)^{2013}(x-4)+...+(x-4)^{2014}+1\right)=0.$$ Now, since the equation $y^{2017}=1$ has unique real root and the equation $y^{2015}=1$ has unique real root, we see that $$y^{2016}-y^{2015}+...+1>0$$ and $$y^{2014}-y^{2013}+...+1>0$$ and we see that can be $x-3=0$, which says that $3$ is an unique root of our equation.

Answer 2

Hint: calculate the derivative of $P$.

Hint2: $P'$ is positive in $\mathbb{R}$.