Ok so I have to find a vertical asymptote with denominator 3x^3-x-1. So then the denominator must not equal zero. But i just can't solve the cubic function i tried long division! but it didn't work
I would graph it and find the roots.
We aren't allowed graphing calculators, plus it a rational function
Finding the vertical asymptote of a function can be a bit tricky, especially if you are dealing with a cubic function like the one you mentioned. Here are the steps you can follow to find the vertical asymptote:
Step 1: Set the denominator equal to zero and solve the equation.
In your case, the denominator is 3x^3 - x - 1. So, set it equal to zero:
3x^3 - x - 1 = 0
Step 2: Try factoring the equation.
Sometimes, the equation can be factored, making it easier to find the roots. However, cubic functions are notoriously difficult to factor, so it's often not possible to factor them nicely.
Step 3: Use numerical methods to solve the equation.
Since finding the roots of a cubic equation can be challenging, you can use numerical methods such as the Newton-Raphson method or the bisection method to approximate the roots. These methods involve iterative calculations and may require the use of a computer program or calculator.
Step 4: Determine the vertical asymptote.
The vertical asymptote occurs when the denominator is equal to zero. The values you found in step 3 will be the x-values at which the function approaches infinity or negative infinity, indicating the presence of vertical asymptotes.
If you are having trouble solving the cubic equation using long division, it may be helpful to try other methods like factoring or using numerical techniques. Keep in mind that finding the exact solution may not always be possible, but estimating the roots can still be useful in finding the vertical asymptote.