# Find all polynomials that satisfy a functional equation [closed]

I need help with this problem if anyone can contribute to solving this I would much appreciate it:

Find all polynomials $$P$$ that satisfy $$P(1)=210$$ and for every $$x\in \mathbb{R}$$: $$(x+10)P(2x)=(8x-32)P(x+6).$$

• Welcome to MSE. What have you tried? Dec 24, 2020 at 20:13
• Plug in $x = \frac{1}{2}$, what do you get? Dec 24, 2020 at 20:24
• Also try $x= -5$
– IanJ
Dec 24, 2020 at 20:29
• Ok thank you all for your contribution Dec 24, 2020 at 20:32

small hint

With $$x=-10$$, we get $$P(-10+6)=P(-4)=0$$ and for $$x=4$$, it gives $$P(8)=0$$

thus

$$P(x)=(x-8)(x+4)R(x)$$ with $$R(1)=\frac{210}{-35}=-6$$

• Subbing $x=2$ gives yet another root. Dec 24, 2020 at 20:38
• Thank you very much Dec 24, 2020 at 20:38

$$P(x) = c (-8 + x) (-4 + x) (4 + x)$$

• My solution is right. Who down voted it? Dec 24, 2020 at 20:27
• I didn't down vote it Dec 24, 2020 at 20:33
• upvote if you find it useful Dec 24, 2020 at 20:33
• at least will even out Dec 24, 2020 at 20:34
• True i forgot to sum +6 f(x-6), now it is updated Dec 24, 2020 at 20:43