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Question 717575: Solve the equation
x^3+4x^2-9x-6=0
giving each root in an exact form
I do not even know how to start this question, sorry!
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! First of all, try to post your questions in an appropriate category. This problem has nothing to do with logarithms. (I have changed the category to an appropriate one.)
To solve an equation like this we want a zero on one side and then we try to factor the other side. Since we already have a zero on one side we can go straight to the factoring. The greatest common factor is 1 (which we rarely bother factoring out). With four terms, there are too many terms for factoring with patterns or for trinomial factoring. I don't see a way to factor by grouping so all we have left is factoring by trial and error of the possible rational roots.
The possible rational roots of a polynomial are all the possible ratios, positive and negative, which can be formed using a factor of the constant term (at the end) over a factor of the leading coefficient (at the front). The constant term of this polynomial is 6. (Actually it is -6 but since we're going to include all positive and negative ratios anyway, it doesn't matter if we use 6 or -6.) The factors of 6 are 1, 2, 3 and 6. The leading coefficient is 1 whose factors are 1's. So the possible rational roots of the left side are:
+1/1, +2/1, +3/1 and +6/1
which simplify to:
+1, +2, +3 and +6
If a number, let's call it "r", is a root of a polynomial then (x-r) is a factor of the polynomial. We can check 1 and -1 mentally since powers of 1 and -1 are simple. Neither of these numbers turn out to be roots.
For the other possible rational roots we are going to use synthetic division to see if (x-r) (where "r" is one of our possible rational roots) is a factor. Staring with 2:
2 | 1 4 -9 -6
---- 2 12 6
-----------------
1 6 3 0
The number in the lower right corner is the remainder. A zero remainder means that (x-2) divided evenly into the polynomial. This also means that (x-2) is a factor of the polynomial. Not only that, the rest of the bottom row tells use what the other factor is. The "1 6 3" translates into 
The left side is now:
Neither factor will factor further. Now we can use the Zero Product Property:
 or 
Solving the first equation is easy, x = 2. To solve the second equation we need to use the Quadratic Formula:
Simplifying...
which is short for:
 or 
So there are three solutions to your equation:
 or or
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