Hi! I was wondering if you guys could offer me help on the following problem. I am supposed to find the x-intercepts of this problem, but all of my attempts have failed. I have tried factoring it ,and I tried the Rational Zero Theorem. Thanks for your help!
2x^4-7x^3+11x-4=0
This polynomial does not factor with rational coefficients, so you will need to use a numeric or graphical method. I'm not sure what tools you have at your disposal, but there are several good graphing calculators online, such as
http://www.wolframalpha.com/input/?i=2x%5E4-7x%5E3%2B11x-4
Thank you so much! I was debating whether or not I should use a graphing calculator for this problem. Is there a way I can determine if the graph touches or crosses at the x-axis without the calculator in this case?
Well, you can evaluate y at several places and you know that if it is positive at x1 and negative at x2, there is at least one root in between them. This can be repeated to the desired precision. You can use Descartes' Rule of Signs to figure how many positive or negative roots there might be. And you can use synthetic division sign checks to bound the range of the real roots.
Thanks!
Of course! I'd be happy to help you find the x-intercepts of the equation 2x^4-7x^3+11x-4=0.
Factoring and the Rational Zero Theorem are both great methods to consider, but in this case, factoring might not be the most efficient approach. Instead, we can use another method called the numerical method or root-finding method, which involves finding approximate solutions using numerical calculations.
One popular numerical method is the Newton-Raphson method. The general idea behind this method is to start with an initial guess for a solution and then iteratively refine it to find a more accurate value. Here's how you can apply the Newton-Raphson method to your equation:
1. Start by making an initial guess for a value of x.
2. Use the derivative of the equation, which is 8x^3 - 21x^2 + 11, to calculate the slope of the curve at that point.
3. Use the initial guess and its corresponding slope to find the equation of the tangent line.
4. Find the x-intercept of the tangent line (where the line intersects the x-axis), which will be a more accurate approximation of the solution.
5. Repeat steps 2 to 4 with the newly found x-intercept as the new initial guess, until you achieve the desired level of accuracy.
Alternatively, you can also use software or online tools such as graphing calculators or equation solvers to find the approximate values of the x-intercepts. These tools use algorithms similar to the Newton-Raphson method to find solutions.
I hope this helps you find the x-intercepts of the equation 2x^4-7x^3+11x-4=0. Let me know if you have any further questions!