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

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Question 188143: 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:

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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