The number of real roots of $x^5 - 5x + 2 =0$

Question

How many real roots does the equation $x^5 - 5x + 2 =0$ have?

I know the following facts:

1. The equation will have odd number of real root.
2. That function cannot have rational root.
3. The function will have two real roots between $(1,2)$ and $(0,1)$.

Can anyone please help me in solving this problem?

Answer 1

Hint. Note that $f'(x)=5(x^4-1)=5(x^2+1)(x-1)(x+1)$ which implies that $f$ is strictly increasing in $(-\infty,-1]$, it is strictly decreasing in $[-1,1]$ and it is strictly increasing in $[1,+\infty)$. Knowing that $\lim_{x\to \pm \infty}f(x)=\pm\infty$, $f(-1)=6$, and $f(1)=-2$, what may we conclude?

Answer 2

Hint:

If you take the derivative twice, you get $x\mapsto20x^3$, which is negative for $x<0$ and positive for $x>0$. This tells you that $f'$ is decreasing on $\Bbb R_-$ and increasing on $\Bbb R_+$. Therefore $f'$ can vanish at most twice.

What would happen to $f'$ if $f(x)=0$ had $5$ solutions?

Answer 3

As you have worked out, there is at least one root in each of the intervals $(0,1)$ and $(1,2)$. The number of positive roots as given by the rule of signs is zero or two, so there are exactly two positive roots. Applying the rule of signs to the polynomial with $x$ replaced by $-x$ ($-x^5+5x+2$) we see there is exactly one negative root. Lastly, $x=0$ is obviously not a root, so there are exactly three real roots of the polynomial.

Answer 4

Using Descartes' Rule of Signs ...

$p(x) = x^5-5x+2$ has two sign differences, so $p(x)$ has either $2$ or $0$ positive zeros.

$p(-y) = -y^5 + 5y + 2$ has one sign difference, so $p(-y)$ has $1$ positive zero, that is $p(x)$ has $1$ negative zero.

From your fact 3, we conclude $p(x)$ has at least $2$ positive zeros.

Result: $p(x)$ has exactly $3$ zeros, two of them positive, one of them negative.

Answer 5

Here is a solution without calculus or Descartes's Rule of Signs. However, some knowledge about continuity of polynomial functions is required.

Let $a,b,c,d,e$ be the roots of this polynomial. Using Vieta's Formulas, we have $$a+b+c+d+e=0$$ and $$ab+ac+ad+ae+bc+bd+be+cd+ce+de=0\,.$$ This means $$\begin{align} a^2&+b^2+c^2+d^2+e^2 \\&=(a+b+c+d+e)^2-2(ab+ac+ad+ae+bc+bd+be+cd+ce+de) \\&=0-2\cdot 0=0\,.\end{align}$$ Since $0$ is not a root of this polynomial, we conclude that not all roots are real (otherwise, it must hold that $a^2+b^2+c^2+d^2+e^2>0$). Thus, the polynomial has either one or three real roots. Since the polynomial has at least one root in each of the three intervals $(-2,-1)$, $(0,1)$, and $(1,2)$, we conclude that there are exactly three real roots.

Answer 6

Just adding to RobertZ's precise answer:

Beyond $1$ and before $-1$, the function is strictly increasing.

Since $f(-1)= 6$ and $f(1)= -2$, from Intermediate value theorem, the function must attain a zero in $(-1,1)$. Similarly it attains a zero in $(1, \infty)$ and $(-\infty , -1)$.

So it has $3$ roots.

Answer 7

We can use Sturm's theorem to find a definitive answer to this (unlike Descartes, it counts exactly how many distinct real roots there are). A Sturm chain for $X^5-5X+2$ is given by $$ \left( X^5-5X+2 , 5X^4 -5 , 4X-2 , \frac{75}{16} \right). $$ The last term is a multiple of the discriminant, and in particular, is not zero, so there are no repeated roots. It suffices to examine the sign changes between the leading coefficients, and subtract from the number of sign changes in the leading coefficients when $X$ is replaced by $-X$. The former chain is $$ (1,5,4,75/16), $$ which has no sign changes, while the latter is $$ (-1,5,-4,75/16), $$ which has three sign changes. Hence there are $3-0=3$ real roots.