Use the Factor Theorem to determine the rational zeros of this function. Then use other methods to determine any other zero of this function.
4x^4-x^3-11x^2-2x+6
(I already did p/q and found a total of 16 possible zeros. I synthetic divide all of them and none gave me a root of zero. What do I do next?)
There must be a typos somewhere if they expect you to find rational zeros. There are none.
Or, they knew that, and just want you to use some numeric methods like bisection, secants, Newton's method, etc.
However, since the problem is in a section discussing the Factor Theorem, I'd expect there to be at least one rational root, so check for a typo.
To determine the rational zeros of a polynomial function, we can use the Rational Root Theorem. The Rational Root Theorem states that if a polynomial function has a rational zero, it will be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Given the function f(x) = 4x^4 - x^3 - 11x^2 - 2x + 6, the constant term is 6 and the leading coefficient is 4. Therefore, the possible rational zeros of the function are given by the factors of the constant term (6) divided by the factors of the leading coefficient (4).
You have already found a total of 16 possible zeros using the p/q method, but none of them gave you a root of zero. If none of the possible rational zeros found using the Rational Root Theorem are a root of the function, it means that there are no rational zeros for this function.
To find any other zeros of the function, you can use other methods like factoring, graphing, or using numerical methods such as the Newton-Raphson method or the bisection method.
Since factoring does not seem like a viable option for this specific function, let's try graphing or numerical methods to find the other zeros.
For graphing, you can plot the function and look for x-values where the graph intersects the x-axis. You can use graphing software or online tools to plot the function and find the approximate zeros.
If you want to use numerical methods, such as the Newton-Raphson method, you need to have an initial guess for the zero. This method uses the derivative of the function to approach the zero. It involves iterating a process until you get a close approximation of the zero.
Another numerical method is the bisection method, which involves iteratively bisecting an interval where the function changes sign until you get an approximation of the zero.
Overall, if the Rational Root Theorem doesn't yield any rational zeros, you can try other methods like graphing or numerical methods to find the other zeros of the function.