Solve cubic with no rational zeros

In summary, the conversation discusses a problem statement of x^3-8x+10=0 and the attempt to solve it by dividing by x-1. The result was (x-1)(x^2+x-7)+3+10=0, which simplifies to x^3-8x+20=0. There is one real solution, but it cannot be easily calculated without knowledge of solving cubics by radicals. A decimal approximation of the solution is -3.318628218.
  • #1
biochem850
51
0

Homework Statement



x^3-8x+10=0

Homework Equations





The Attempt at a Solution



By dividing by x-1 (1 is a factor of the solution), I got (x-1)(x2+x-7)+3+10=0

which equals x^3-8x+20=0

I think the solution would be imaginary but I used a graphing calculator and there is one solution but I don't know how to calculate it.
 
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  • #2
biochem850 said:

Homework Statement



x^3-8x+10=0
What is the complete problem statement?
biochem850 said:

Homework Equations





The Attempt at a Solution



By dividing by x-1 (1 is a factor of the solution), I got (x-1)(x2+x-7)+3+10=0
x - 1 is NOT a factor of x^3 - 8x + 10.
biochem850 said:
which equals x^3-8x+20=0

I think the solution would be imaginary but I used a graphing calculator and there is one solution but I don't know how to calculate it.

We can't help you if we don't know what it is that you're supposed to do. That's where the complete problem statement would be useful information.
 
  • #3
biochem850 said:

Homework Statement



x^3-8x+10=0

Homework Equations





The Attempt at a Solution



By dividing by x-1 (1 is a factor of the solution), I got (x-1)(x2+x-7)+3+10=0

which equals x^3-8x+20=0

I think the solution would be imaginary but I used a graphing calculator and there is one solution but I don't know how to calculate it.

Unless you know how to solve a cubic by radicals, you won't be finding the exact answer for the real root. Maple gives it as:

-(1/3)*(135+3*sqrt(489))^(1/3) - 8/(135+3*sqrt(489))^(1/3)

and a decimal approximation to that is -3.318628218.
 

1. How do I identify a cubic equation with no rational zeros?

A cubic equation with no rational zeros is one in which all of the coefficients (including the constant term) are integers and there are no common factors among them. Additionally, the discriminant of the equation (b^2-4ac) must be negative, indicating that there are no real solutions.

2. Can a cubic equation with no rational zeros have any real solutions?

No, a cubic equation with no rational zeros only has complex solutions. This means that the solutions involve imaginary numbers, which can be represented as a combination of real numbers and the imaginary unit, i.

3. How do I solve a cubic equation with no rational zeros?

The most common method for solving a cubic equation with no rational zeros is to use the cubic formula, also known as the Cardano's formula. This involves finding the roots of the equation using complex numbers and then simplifying the expression to get the final solutions.

4. Are there any other methods for solving a cubic equation with no rational zeros?

Yes, there are other methods such as using the cubic graphing calculator or using numerical methods such as Newton's method or the bisection method. However, these methods may not always give exact solutions and may require multiple iterations to get closer to the actual solutions.

5. What are some real-life applications of solving cubic equations with no rational zeros?

Cubic equations with no rational zeros are often used in physics and engineering to model and solve real-world problems involving complex systems. They are also used in fields such as economics and finance to analyze and predict market trends and behavior.

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