# Does there always exists coefficients $c,d\in\mathbb{R}$ s.t. $ax^3+bx^2+cx+d$ has three different real roots?

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?

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.
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.