Thread: How do you find all real ratios..for this eq..[step by step]

f(x)=x^5+9x^4+13x^3-39x^2-56x-12

Originally Posted by Valaina

f(x)=x^5+9x^4+13x^3-39x^2-56x-12

Do you mean all real roots? If so I would suggest the factor theorem.

Start with the Rational Zeros Test, which states that all potential rational zeros of a function is in the form of P/Q, where P is a factor of the constant term, and Q is a factor of the leading coefficient. The potential rational zeros of

would be


Now, start testing the possible roots using synthetic division. -12 doesn't work (needs to give a remainder of 0), but -6 does:

Code:

-6| 1  9  13 -39 -56 -12
--    -6 -18  30  54  12
------------------------
    1  3  -5  -9  -2   0

Continue to try the potential zeros, but now on the quotient of the above division, which is -4, -3, and -2 don't work, but -1 does:

Code:

-1| 1  3  -5  -9 -2
--    -1  -2   7  2
-------------------
    1  2  -7  -2  0

You're now left with the polynomial Using this as the new dividend, 1 doesn't work, but 2 does:

Code:

2| 1  2  -7  -2
-     2   8   2
---------------
   1  4   1   0

So far we've found 3 roots: -6, -1, and 2. The factorization so far is
To find the remaining two roots, solve the quadratic . This can't be factored evenly, so either complete the square or use the quadratic formula.


Edit: hit Submit too soon...


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