I am trying to find the number of real roots of the polynomial $\mathrm{f(x)=x^5-5x+3}$, using the Sturm theorem. I have started writing out the Sturm sequence:
- $\mathrm{p_0=x^5-5x+3}$
- $\mathrm{p_1=5x^4-5}$
- $\mathrm{p_2=4x-3}$
- $\mathrm{p_3= 5-\frac{15}{4}x^3}$
However, this is where I am stuck. I have no idea how I will divide $\mathrm{p_2}$ by $\mathrm{p_3}$ to find the remainder. How do I proceed from this point?
The polynomial $p_3$ is the opposite of the remainder of the division of $p_1=5x^4-5$ by $p_2=4x-3$. Since $p_2$ has degree $1$, it follows that $p_3$ is a constant: $$p_3(x)=-p_1(3/4)=-5((3/4)^4-1)>0.$$ Then the sequence of signs at $-\infty$ is $-+-+$ which has three sign changes. At $+\infty$ the sequence is $++++$ which has no sign change. Hence the number of roots of the given polynomial $x^5-5x+3$ is $3-0=3$.