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

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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).


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Now simply use synthetic division to find the real rational zeros


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :

1	|	1	-2	3	-6
	|		1	-1	2
		1	-1	2	-4

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


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Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :

2	|	1	-2	3	-6
	|		2	0	6
		1	0	3	0

Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of



Because is a zero, this means that is a factor of

The first three numbers in the last row 1, 0, and 3 form the coefficients to the polynomial . So this consequently means that


Set the right side equal to zero


or Set each factor equal to zero


Since we know that is already a zero, we can ignore the first equation. So simply solve the quadratic equation to find the remaining solutions:


Start with the given equation


Subtract 3 from both sides


or Take the square root of both sides (don't forget the "plus/minus")


or Simplify. Note:




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Answer:


So the solutions of are , or