How can I find the roots of a cubic equation?

In summary, James has been looking at practice exams for the University he wants to apply to and found that he has to take the exam on November 4th. He is struggling with finding the roots of a cubic equation and the formulas are difficult to understand. He has been advised to focus on finding linear and non-factorable quadratic factors and equating the function to zero. He may also need to use Rational Roots Theorem and polynomial division, but he may need an example to fully understand. He is determined to learn this material and improve his knowledge for university.
  • #1
James...
25
0
I've been looking at some practise exams for the University I would like to apply to, I have to sit the exam on 4th November.

We have never done finding the roots of a cubic equation before and I cannot figure it out from looking on the internet, the formulas are all horrible to understand.

Any help would be appreciated.

James
 
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  • #2
Knowing many formulas does not help. What helps is finding the linear binomial factors and any nonfactorable quadratic factors; and equate the whole expression to zero. You are interested in where the function equals zero, or where the function shares a point with the x-axis.

Your wish for knowing "formulas which are horrible to understand" reflects ones first-through study of College Algebra, in which you learn to deal with Descartes Law of Signs, Rational Roots Theorem, the Factor and Remainder Theorems. You could spend a good 2 months studying this stuff from a College Algebra textbook and become very well knowledgeable about these things. You really want to KNOW this stuff and you are not really concerned with a course grade for credit. Those laws and theorems really CAN be well learned if you spend the time seriously studying them; keeping the knowledge several weeks later is another thing entirely.
 
  • #3
(CONTINUED)...
Actually for just a cubic equation, you don't need those fancy formulas. Usually, you would be expected to factor into linear and possibly a quadratic factor and determine what values of the variable make the product equal to zero.
 
  • #4
(one more thought)
You could resort to Rational Roots Theorem and polynomial division, but not much else would be needed. Do you need an example?
 
  • #5
Many many thanks for your replies, I think I will try and have a go at studying the things you mentioned in your first post for definate! Anything that will expand my knowledge and help me at University can only be a good thing!

I managed to get the answer by creating a linear and quadratic factor of the cubic equation. We have done this before many times, but never as part of a polynomial, I should have realized I could have done it as it equals zero and therefore I can split it into factors.

I will have a play about with the Rational Roots theorum and polynomial division tomorrow but I may need an example depending on how i find it.

Thanks again
James
 

1. What is the cubic equation?

The cubic equation is a polynomial equation of the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and x is the variable. It can have up to three solutions or roots.

2. How do you find the roots of a cubic equation?

There are several methods for finding the roots of a cubic equation, including factoring, using the rational root theorem, and using the cubic formula. The most common method is using the cubic formula, which involves plugging in the values of a, b, c, and d into a formula to find the roots.

3. Are there any special cases for finding the roots of a cubic equation?

Yes, there are three special cases that can occur when finding the roots of a cubic equation. The first case is when all three roots are real and distinct. The second case is when two roots are complex and one is real. The third case is when all three roots are real, but two are equal.

4. Can you find the roots of a cubic equation without using the cubic formula?

Yes, it is possible to find the roots of a cubic equation without using the cubic formula. However, this method can be more complex and may require advanced mathematical techniques. It is generally recommended to use the cubic formula or other established methods for finding the roots.

5. Why are the roots of a cubic equation important?

The roots of a cubic equation are important for solving a variety of mathematical and scientific problems. They can be used to determine the maximum and minimum points of a cubic function, find the intersection points of two cubic functions, and solve real-life problems related to rates, distances, and volumes.

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