Question 501940: what are the zeros of this polynomial function and all the work:
f(x)=x^4-4x^3-20x^2+48x
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Factor out an 
\ =\ 0)
If a rational zero exists for a polynomial with integer coefficients, then that rational zero will be of the form  where the constant term is evenly divisible by  and the lead coefficient is evenly divisible by 
Hence, the possible rational zeros for the cubic factor above are: 
Use synthetic division to test these possible zeros, one by one, until you either find one that works or you have exhausted the list. Hint: Start with the low numbers. In the first place, the arithmetic is easier and secondly I have it on good authority that you will find one that works much more quickly that way. Also, if you have a graphing calculator, you can run a quick graph to find an approximation of where the roots ought to lie giving you a hint as to your strategy for picking trial divisors.
If you need a refresher on synthetic division, then go to: Purple Math Synthetic Division (Note that there are 4 pages to read)
Once you have found a trial divisor that works, i.e. you are left with a zero remainder, then you can formulate the next factor:  where  is the successful trial divisor.
The quotient you derive from the synthetic division process will be an easily factorable quadratic. (-6 plus 4 is -2 and -6 times 4 is -24)
Once you have identified all four factors, set each one equal to zero (remember the zero product rule?) and solve to find each of the four roots.
John
My calculator said it, I believe it, that settles it
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