Real root of an equation

Problem. Find the real root of the equation f(x)=x^{3}-2x^{2}+3x-5=0

with an accuracy upto 10^{-4}. Use the method of chords.


Let us first make sure that the given equation has only one real root. This follows from the fact that the derivative

f^{'}(x)=3x^{2}-4x+3>0 (use your knowledge of theory of equations here!!!)

Then, from f(1)=-3<0 and f(2)=1>0, it follows that the given polynomial has a single positive root, which lies in the interval (1,2).

Using the method of chords, we obtain the first approximation:


Since f(1.75)=-0.5156<0 and f(2)=1>0, then 1.75<\xi<2.

The second approximation:

x_{2}=1.75+(0.5156/1.5156).o.25=1.8350 and

since f(1.835)=-0.05059<0, then 1.835<\xi<2.

The sequence of approximations converges very slowly. Let us try to narrow down the interval, taking into account that the value of the function f(x) at the point x_{2}=1.835 is considerably less in absolute value than f(2). We have


Hence, 1.835<\xi<1.9.

Applying the method of chords to the interval (1.835,1.9), we will get a new approximation:


Further calculations by the method of chords yield

x_{4}=1.8437 and x_{5}=1.8438

and since f(1.8437)<0 and f(1.8438)>0, then \xi \approx 1.8438 with required accuracy of 10^{-4}.

You can also solve this problem by the method of tangents. Try it!

More later,

Nalin Pithwa

Fifth degree polynomial equation

Question: Prove that the equation 3x^{5}+15x-8 has only one real root.


The existence of at least one real root follows from the fact that the above polynomial is an odd power.

Let us prove the uniqueness of such a root by contradiction.

Suppose there exist two roots x_{1}<x_{2}. Then, in the interval [x_{1},x_{2}] the given polynomial function satisfies all the conditions of Rolle’s Theorem: it is continuous, it vanishes at the end points and has derivative at all points. Consequently, at some point \xi such that x_{1}<\xi<x_{2},

f^{'}(\xi) equals zero. But, f^{'}(x)=15(x^{4}+1)>0. But, this contradicts hypothesis. Hence, the proof.

More later,

Nalin Pithwa