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Prove that $(x-1)^3+(x-2)^3+(x-3)^3+(x-4)^3=0$ has only one real root. It's easy to show that the equation has a real root using Rolle's theorem. But how to show that the real root is unique? By Descartes' rule of sign, it can be shown that it has 3 or 1 real root.

But it doesn't guarantee that the real root is unique. If we calculate the root then it can be shown that it has only one real root.

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    $\begingroup$ The function is strictly increasing. If $x_1 > x_0$ then $x_1 - c > x_0 -c$ and $(x_1 -c)^3 > (x_0 -c)^3$ and $\sum (x_1 -c)^3 > \sum (x_0 - c)^3$. $\endgroup$ – fleablood Nov 17 at 16:05
  • $\begingroup$ The real root also has multiplicity 1. $\endgroup$ – UtilityMaximiser Nov 18 at 1:19
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A solution that only uses school algebra:

Substitute $t=x-5/2$. Then the equation becomes $$(t+3/2)^3+(t+1/2)^3+(t-1/2)^3+(t-3/2)^3=0.$$ Expanding the brackets, we get $$4t^3+15t=0,$$ or $t(4t^2+15)=0$, which clearly only has one real root.

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    $\begingroup$ Good and simple answer so have my +1, but do you have any intuition why one should use the transformation t=x-5/2 and not something else? $\endgroup$ – Rebel-Scum Nov 18 at 9:13
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    $\begingroup$ @Rebel-Scum To exploit the symmetry of the four terms around $x-5/2$. 5/2 is the average of 1, 2, 3, 4. $\endgroup$ – Evangelos Bampas Nov 18 at 10:22
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    $\begingroup$ @EvangelosBampas Indeed, but that is not mentioned in the answer and the transformation comes a bit out of the blue $\endgroup$ – Rebel-Scum Nov 18 at 11:22
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The function is strictly increasing so the function is one to one.

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Hint:

It is the sum of four increasing functions, and $\lim_{x\to-\infty}=-\infty$, $\lim_{x\to+\infty}=+\infty$. The intermediate value theorem guarantees there is a root, and monotonicity ensures there can be no more than one.

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Here is a rather primitive solution (although judging by the contents of your question you may already know this):

By symmetry, we can see that $x=2.5$ is a real root. We want to show that it is the only real root. Expanding the binomial cubes, we get $$4x^3-30x^2+90x-100$$ which factors into $$2(2x-5)(x^2-5x+10).$$

As the discriminant of the quadratic factor is negative, this tells us that $x=2.5$ is the only real root.

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Hint:

Can you show the derivative is positive for all $x$?

Thus, the function is strictly increasing for all $x$.

$x=2.5$ is a root.

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    $\begingroup$ Is the derivative really necessary here? $\endgroup$ – Bernard Nov 17 at 16:10
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    $\begingroup$ I suppose it is one way to show that a function is strictly increasing. $\endgroup$ – Andrew Chin Nov 17 at 16:11
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    $\begingroup$ and a fairly straightforward way in this case, though it's not really necessary $\endgroup$ – J. W. Tanner Nov 17 at 16:28
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Let $y=\dfrac{x-1+x-2+x-3+x-4}4$

$x=y+2.5$

$$(x-1)^3+(x-2)^3+(x-3)^3+(x-4)^3=(y+1.5)^3+(y+.5)^3+(y-.5)^3+(y-1.5)^3$$

Now use $(a-b)^3+(a+b)^3=2(a^3+3ab^2)$

$$0=4y^3+6y((1.5)^2+(.5)^2)$$

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The function $x\mapsto x^3$ is increasing, since $x^3-y^3=(x-y)(x^2+xy+y^2)$ and $x^2+xy+y^2=\frac{(x+y)^2}{2}+\frac{x^2+y^2}{2}\geq 0$.

Shifting the domain of an increasing function does not change the fact that it is increasing, so also $x\mapsto (x-n)^3$ is increasing for all $n$.

Adding increasing functions results in another increasing function.

Thus, for all natural numbers $n$ the polynomial $$ \sum_{i=1}^n (x-i)^3 $$ is increasing. Since it tends to $\infty$ as $x\to\infty$ and $-\infty$ as $x\to-\infty$, it follows that the polynomial has exactly one value at which it crosses the $x$-axis, that is, exactly one real root.

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Here is an elementary way that uses only

  • $(1)$: $a^3+b^3 = (a+b)(a^2-ab+b^2)$ and
  • $(2)$: $a^2+b^2 >ab$ for $|a|+|b| > 0$

Note, that $|x-1|+|x-4|\geq |x-1 +(4-x)| =3 > 0$ and $|x-2|+|x-3|\geq |x-2 +(3-x)| =1 > 0$

Let's call $p(x) = (x-1)^3+ (x-2)^3 + (x-3)^3 + (x-4)^3$.

Now, using $(1)$ write $$(x-1)^3 + (x-4)^3 = (x-1 + x-4)\left((x-1)^2 + (x-4)^2 - (x-1)(x-4)\right)$$ $$= (2x-5)\left((x-1)^2 + (x-4)^2 - (x-1)(x-4)\right)$$ $$(x-2)^3 + (x-3)^3 = (x-2 + x-3)\left((x-2)^2 + (x-3)^2 - (x-2)(x-3)\right)$$ $$= (2x-5)\left((x-2)^2 + (x-3)^2 - (x-2)(x-3)\right)$$

Hence,

$$p(x) = $$ $$(2x-5)\color{blue}{\left(\underbrace{(x-1)^2 + (x-4)^2 - (x-1)(x-4)}_{\stackrel{(2)}{>}0} + \underbrace{(x-2)^2 + (x-3)^2 - (x-2)(x-3)}_{\stackrel{(2)}{>}0}\right)} $$

So,

$$p(x) = (2x-5)\color{blue}{q(x)} \mbox{ with } \color{blue}{q(x)} > 0 \mbox{ for all } x \in \mathbb{R}$$

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