calculus - Prove that polynomial of degree $4$ with real roots cannot have $pm 1$ as coefficients (IITJEE)

So I was going through my 11th class package on Quadratic equations and I saw a question to prove that a polynomial of $4$th degree with all real roots cannot have $\pm 1$ as all its coefficients.

I tried proving it using calculus, by showing that at least one consecutive maxima and minima will lie either above or below the x axis, but couldn't solve it using that.

I also tried using Descartes Rule of Signs but couldn't solve it with that too. Any help?

Answer

Let $f(x)$ be any quartic polynomial with coefficients from $\{ -1, +1 \}$. Replacing $f(x)$ by $-f(x)$ if necessary, we can assume $f(x)$ is monic. i.e.

$$f(x) = x^4 + ax^3 + bx^2 + cx + d\quad\text{ with }\quad a,b,c,d \in \{ -1, +1 \}$$

If $f(x)$ has $4$ real roots $\lambda_1,\lambda_2,\lambda_3,\lambda_4$, then by Vieta's formula, we have

$$\sum_{i=1}^4 \lambda_i = -a, \sum_{1\le i < j\le 4} \lambda_i\lambda_j = b \quad\text{ and }\quad\prod_{i=1}^4 \lambda_i = d$$ Notice $$\sum_{i=1}^4 \lambda_i^2 = \left(\sum_{i=1}^4\lambda_i\right)^2 - 2\sum_{1\le i < j \le 4}\lambda_i\lambda_j = a^2 - 2b = 1 -2b$$

Since $\sum_{i=1}^4 \lambda_i^2 \ge 0$, we need $b = -1$. As a result,

$$\sum_{i=1}^4 \lambda_i^2 = 3$$ By AM $\ge$ GM, this leads to

$$\frac34 = \frac14\sum_{i=1}^4 \lambda_i^2 \ge \left(\prod_{i=1}^4 \lambda_i^2\right)^{1/4} = (d^2)^{1/4} = 1$$ This is impossible and hence $f(x)$ cannot has 4 real roots.