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Let $P(x) = x^n + ax+b$ with $n\geq 2$ and $a,b\in\mathbb{R}$. Then one of these is true:

  1. the number of distinct real roots of $P$ can be any number between $0$ and $n$.
  2. the number of distinct real roots of $P$ is less than or equal to $3$.
  3. the number of distinct real roots of $P$ is at least $1$.
  4. $P$ cannot have multiple roots.
  5. $P$ has always at least one multiple root if $n\geq 3$.

Is easy to notice that 3. and 4. are false, but I cannot decide between 1.,2. and 5.

Any hint?

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  • $\begingroup$ Hint: Try ofr $n$ even and $n$ odd. (test: it should give you answer number 2.) $\endgroup$ Commented Oct 26, 2013 at 1:14
  • $\begingroup$ $P(x)=x^n+ax+b=0\iff x=\sqrt[n]{-b-a\sqrt[n]{-b-a\sqrt[n]{-b-a\sqrt[n]{-b-\ldots}}}}$ $\endgroup$
    – Lucian
    Commented Oct 26, 2013 at 1:39

2 Answers 2

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I think 5 is false. For example, $x^{3}+x=x(x^2+1)$ has roots 0,i, -i and no multiple root.

May be statement 2 is correct.

  1. If a is not equal $0$

$x^n + ax + b = 0$

$x^n = -ax -b$

Graph functions $y = x^n$ and line $y= -ax - b$.

X-coordinates of their intersections would be real roots of P.

Maximum number of intersections is 3 if n is odd.

If n is even maximum number of intersections is 2.

Statement 2 is true.

  1. If $a=0$

$x^n= - b$

There are $0$ or $1$ or $2$ distinct real roots.

So statement 2 is true and statement 5 is false.

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Hint 1 $a=0, b=-1$ takes care of $5$.

Hint 2: How many distinct roots does $f''$ have? How many roots can $f'$ have? What about $f$?

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