- #1

- 870

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## Homework Statement

"Show that for some ##x\in ℝ##, that ##x^5+2x^4+3x^3+2x^2+x=1##."

## Homework Equations

## The Attempt at a Solution

Okay, so I know from Descartes' rule of sign that the function ##f(x)=x^5+2x^4+3x^3+2x^2+x-1## has exactly one positive root, since the sign of the coefficients change exactly once in the entire function. But I am asked to show that it has a positive root, and I am not fully sure that this explanation would cut it.

I also have that ##x^5+2x^4+3x^3+2x^2+x=\frac{1}{2}x^5+2x^4+3x^3+2x^2+\frac{1}{2}x+\frac{1}{2}x(1+x^4)=\frac{1}{2}x\sum_{k=0}^4 \binom 4 k x^k+\frac{1}{2}x(1+x^4)=\frac{1}{2}x[(1+x)^4+(1+x^4)]##. But I have to show that this expression is identically one, and I haven't an idea of how to do this.