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Question 202826: Which of the following is not a possible rational zero of P(x)=6X^3-2X^2-19X-3?
Found 2 solutions by stanbon, jsmallt9:
Answer by stanbon(75887)   (Show Source):
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website! 
The possible rational roots of any polynomial function can be listed by creating all possible fractions, both positive and negative, where the numerator is one of the factors of the constant term (at the end) and the denominator is one of the factors of the leading coefficient.
In your problem the constant term is -3 whose factors are 1, -3, -1 and 3.
In your problem the leading coefficient is 6 whose factors are 1, 6, -1, -6, 2, 3, -2 and -3.
The possible rational roots of P(x) are all the possible fractions, positive and negative, that can be formed with 1, -3, -1 or 3 in the numerator and with 1, 6, -1, -6, 2, 3, -2 or -3 in the denominator:
1/1 = 1
1/6
1/-1 = -1
1/-6 = -6
1/2
1/3
1/-2 = -(1/2)
1/-3 = -(1/3)
-3/1 = -3
-3/6 = -(1/2) (already listed)
-3/-1 = 3
-3/-6 = 1/2 (already listed)
-3/2 = -(3/2)
-3/3 = -1 (already listed)
-3/-2 = 3/2
-3/-3 = 1 (already listed)
The rest of the possible fractions, with -1 and 3 in the numerator, only result in fractions we have already found. So the above list, without the duplicates, is the list of all possible rational roots of P(x).
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