Question 570933: how do i answer the question: list all possible rational zeros of each equation, with equations like x^3-7x+6=0? i have worked out the equation using synthetic division, but that takes too long, is there a faster way?
Answer by Edwin McCravy(20060)   (Show Source):
You can put this solution on YOUR website!
If you're only asked to list all POSSIBLE rational zeros,
then they are NOT asking you to find the zeros.
They are merely asking you to list all the CANDIDATES for
rational zeros.
It is unfortunate that all the algebra books use the word
"POSSIBLE" and ask for all "possible rational zeros". But
they do! They should say "candidates for rational zeros",
because many times the candidates are not really "possible".
If a polynomial has rational zeros, then those rational zeros
will be fractions of this form:
However, not all fractions of this form are necessarily zeros of the
polynomial. Indeed it may happen that none of the fractions so formed
are actually zeros of the polynomial.
Since the leading coefficient is 1, you need not worry about denominators
in this polynomial. All they want you to do is list all the positive and
negative factors of the last term, 6, which are
±1, ±2, ±3, ±6
The only thing they are asking you to do is to list those!
Since you have already found all the zeros, you know that they are
-3, 1 and 2. So you see that +3,-1,-2,+6, and -6 are not zeros at all.
However they were CANDIDATES for aeros and so you should list all
the candidates for solutions.
I think you were misled, and understandably so, by the word "POSSIBLE".
I wish they would use another word.
Edwin
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