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

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

Answer by jim_thompson5910(35256) (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).

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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
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