SOLUTION: Solve: x^3+3x-2x^2-6=0

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Solve: x^3+3x-2x^2-6=0       Log On


   



Question 188143: Solve: x^3+3x-2x^2-6=0


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Any possible rational zero can be found through this formula

where p and q are the factors of the last and first coefficients


So let's list the factors of -6 (the last coefficient):



Now let's list the factors of 1 (the first coefficient):



Now let's divide each factor of the last coefficient by each factor of the first coefficient









Now simplify

These are all the distinct possible rational zeros of the function.




Note: these are the possible zeros. The function may not even have rational zeros (they may be irrational or complex).


------------------------------------------------------------------------------------------------------


Now simply use synthetic division to find the real rational zeros


Let's see if the possible zero 1 is really a root for the function x%5E3-2x%5E2%2B3x-6


So let's make the synthetic division table for the function x%5E3-2x%5E2%2B3x-6 given the possible zero 1:
1|1-23-6
| 1-12
1-12-4

Since the remainder -4 (the right most entry in the last row) is not equal to zero, this means that 1 is not a zero of x%5E3-2x%5E2%2B3x-6


------------------------------------------------------


Let's see if the possible zero 2 is really a root for the function x%5E3-2x%5E2%2B3x-6


So let's make the synthetic division table for the function x%5E3-2x%5E2%2B3x-6 given the possible zero 2:
2|1-23-6
| 206
1030

Since the remainder 0 (the right most entry in the last row) is equal to zero, this means that 2 is a zero of x%5E3-2x%5E2%2B3x-6



Because x=2 is a zero, this means that x-2 is a factor of x%5E3-2x%5E2%2B3x-6

The first three numbers in the last row 1, 0, and 3 form the coefficients to the polynomial x%5E2%2B3. So this consequently means that x%5E3-2x%5E2%2B3x-6=%28x-2%29%28x%5E2%2B3%29


%28x-2%29%28x%5E2%2B3%29=0 Set the right side equal to zero


x-2=0 or x%5E2%2B3=0 Set each factor equal to zero


Since we know that x=2 is already a zero, we can ignore the first equation. So simply solve the quadratic equation x%5E2%2B3=0 to find the remaining solutions:


x%5E2%2B3=0 Start with the given equation


x%5E2=-3 Subtract 3 from both sides


x=sqrt%28-3%29 or x=-sqrt%28-3%29 Take the square root of both sides (don't forget the "plus/minus")


x=sqrt%283%29%2Ai or x=-sqrt%283%29%2Ai Simplify. Note: i=sqrt%28-1%29




==================================================================================================

Answer:


So the solutions of x%5E3-2x%5E2%2B3x-6 are x=2, x=sqrt%283%29%2Ai or x=-sqrt%283%29%2Ai