x^3-9x+1=0
I assume you want to find the roots of the function. Since there are no rational roots, wolframalpha is your friend:
http://www.wolframalpha.com/input/?i=x^3-9x%2B1
It doesn't show you the steps though?
To find the solutions to the equation x^3 - 9x + 1 = 0, we can use either numerical methods or algebraic methods. Let's start with an algebraic method called the Rational Root Theorem.
The Rational Root Theorem states that if a rational number p/q (in simplest form) is a root of a polynomial equation with integer coefficients, then p must be a factor of the constant term (in this case, 1) and q must be a factor of the leading coefficient (in this case, 1). So, we need to check all the possible factor pairs of 1 to find any potential rational roots.
The factor pairs of 1 are (+/-1, +/-1). Therefore, the potential rational roots are +/-1.
We can now use a method like synthetic division or long division to check if these potential roots are actual solutions to the equation. Synthetic division is usually faster, so let's go with that.
Synthetic Division for x^3 - 9x + 1 = 0 with a potential root of 1:
1 | 1 0 -9 1
| 1 1 -8
--------------
1 1 -8 -7
The remainder in the last row is -7, not equal to zero, so 1 is not a root of the equation.
Synthetic Division for x^3 - 9x + 1 = 0 with a potential root of -1:
-1 | 1 0 -9 1
| -1 1 8
--------------
1 -1 -8 9
The remainder in the last row is 9, not equal to zero, so -1 is not a root of the equation.
At this point, we have checked all the possible rational roots, and none of them are roots of the equation. This means that the equation x^3 - 9x + 1 = 0 does not have any rational solutions. However, it is still possible that there are irrational or complex solutions.
To find the remaining solutions, we can use numerical methods like graphing the equation to find its intersections with the x-axis or using numerical approximation methods like Newton's method or bisection method.
Using a graphing calculator or software, we can plot the equation and visually identify the approximate locations of the solutions. In this case, we see that there is one real root between -2 and -1, another real root between 0 and 1, and a third real root between 2 and 3. The exact values of these roots can be approximated using numerical methods.
To summarize, the equation x^3 - 9x + 1 = 0 does not have any rational solutions, and to find the approximate values of the solutions, we would need to use numerical methods like graphing or approximation algorithms.