Can't figure out how.. (Pre-Calc, Rational Zero Theorem and Synthetic Division

I've encountered the problem:

x^3 - 10x - 12

I am suppose to;
a.) list all possible rational roots
b.) use synthetic division to test
c.) find an actual root and use the quotient I found in part "b" to find the remaining roots and solve

So here's what happened:
1. I found all of the possible rational zeros
2. I used synthetic division on each possible root, none of which equaled zero.
3. For part "b" I assumed that there is no zero for this problem...
4. I check the answer in the back of my textbook and found the answer to "b" :
b.) -2
and c.) {-2, 1 + √7, 1 - √7}

I've done all I possibly can with what I know and I couldn't come up with the same answer in the textbook..

Since the leading coefficient is 1, the list of possible rational roots will be integral:



Now, trying -2, we find:



So, we know is a factor. Performing the division, we find:

and so:



We know is one root, the other two are found from the roots of the quadratic factor:

Quote:

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Originally Posted by MarkFL2

Now, trying -2, we find:



So, we know is a factor. Performing the division, we find:

and so:

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What theorem is that? I think I missed the lecture over that theorem...

I'm not sure if you saw my response, (I'm new at the moment, my apologies.)

If r is a root of f(x), then f(r) = 0.

Oh.... alright...
That clears it up! Many thanks!