SOLUTION: Solve: x^3+3x-2x^2-6=0
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Question 188143
:
Solve: x^3+3x-2x^2-6=0
Answer by
jim_thompson5910(35256)
(
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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).
------------------------------------------------------------------------------------------------------
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
------------------------------------------------------
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