# Simplest Way of finding zeros of cubic eq when rational roots test fails

## Homework Statement

Find the zeros of the cubic equation:
$$y = x^3 -9x^2 + 15x + 30$$

How do we find the zeros of this? In this case, subbing in x-values that will make it equal 0 does not work.

Related Calculus and Beyond Homework Help News on Phys.org
Char. Limit
Gold Member
Well, for this one, you're either going to have to solve the (extremely complicated) cubic equation or use Newton's Method, because there isn't an integer or rational number that will solve this equation for y=0.

vela
Staff Emeritus
Homework Helper
Plotting the function is a good idea as well. At the least, you can obtain initial guesses to use with Newton's method.

gb7nash
Homework Helper
I would probably suggest using Newton's method (or bisection method if you can find a positive function value and a negative function value) Once you've found the first root, take f(x)/(x-r), where r is the root. This will roughly yield a quadratic and you can use the quadratic formula to find the remaining two roots.

Char. Limit
Gold Member
I would probably suggest using Newton's method (or bisection method if you can find a positive function value and a negative function value) Once you've found the first root, take f(x)/(x-r), where r is the root. This will yield a quadratic and you can use the quadratic formula to find the remaining two roots.
If he's restricting himself to real numbers, there ARE no other roots. Just the one.

gb7nash
Homework Helper
I'm missing something here. How do we know there's only one real root?

Char. Limit
Gold Member
Oh, sorry. I graphed it first, to get an impression of where the roots were.

gb7nash
Homework Helper
Cheater. :tongue:

Char. Limit
Gold Member
Real mathematicians use pictures!

Dick
Homework Helper
I'm missing something here. How do we know there's only one real root?
You can also conclude there is only one root by looking at the derivative and finding the extreme values plus knowing the behavior as x->+/-infinity. Which is basically 'graphing it' without a calculator. Hence, not cheating.

gb7nash
Homework Helper
You can also conclude there is only one root by looking at the derivative and finding the extreme values plus knowing the behavior as x->+/-infinity. Which is basically 'graphing it' without a calculator. Hence, not cheating.
Good call. This is a good method.

Uh.. That was the first time I ever heard any of those methods, so could anyone give me a few pointers just to show how I start off with one of the methods above?

Char. Limit
Gold Member
Well, Newton's method is given as such: given a guess x0 as to where the root of a function f(x) is, then...

$$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$$

x1 is a better guess, and x2 will be a better guess than x1, and so on.

NOTE: With an initial guess close to the answer, you can get a very good approximation after only two or three iterations of this. For example, in this problem, an initial guess of x0=-1 will get you a very good approximation in three iterations.

Last edited:
Dick