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Well, my question is exactly the one which is written in the title. Consider $a,b\in\mathbb{R}$ two given real numbers. Now, let $c,d\in\mathbb{R}$ be two parameters that we can control. My question is, does there always exists $c,d$ so that the polynomia $$ ax^3+bx^2+cx+d $$ has three different real roots?

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2 Answers 2

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If $a=0$, then there cannot be $3$ real roots.

If $a\not = 0$, WLOG $a\geq0$, then let $c<0$, $d=0$. It passes $(0,0)$ with negative slope, so it has $3$ real roots.

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Let $f(x)= ax^3 + bx^2 + cx + d $. Clearly for $a=0 $ ,$f(x)=0$ cannot have $3$ roots. Let $a\neq0$. Note that only the sum of the roots =$-b/a$ is given as $a$ and $b$ are given. The product of the roots taking $2$ at a time($c/a$) and product of all roots($-d/a$) are not given as $c$ and $d$ are not given. Let the roots be $p-r, p , p+r$ ( in AP $r\neq0$ to make the roots distinct; here $p, r$ are real). Therefore $p-r + p + p+r= -b/a= 3p$. Hence $p=-b/3a$. Therefore the roots becomes $-b/3a-r, -b/3a , -b/3a +r$ . Note that sum of this AP is independent of the value of $r$. Hence we can adjust the value of $c$ and $d$ by finding the product of the roots taking $2$ at a time * $a$ and product of all roots*d respectively. Therefore thre always exists coefficients $c$ and $d$ given $a, b$ such that all roots are distinct and real.

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