What are the possible rational roots of #6x^4+35x^3-x^2-7x-1=0# and then determine the rational roots?

1 Answer
Jan 9, 2017

Answer:

The possible rational roots are #+-1;+-1/2;+-1/3;+-1/6#;
The rational roots are #1/2;-1/3#

Explanation:

The possible rational roots are obtained by dividing each of the factors of the known term (-1) by each of the factors of the maximum degree term's coefficient (6).

Then they are:

#+-1;+-1/2;+-1/3;+-1/6#

To determine the rational roots, let's use the remainder rule:

#color(red)(x=1)-> 6+35-1-7-1=32!=0#
#color(red)(x=-1)-> 6-35-1+7-1=-24!=0#
#color(red)(x=1/2): 6(1/2)^4+35(1/2)^3-(1/2)^2-7(1/2)-1=0#

Then #x=1/2# is a rational root.

You also can use a spreadsheet to find zeros faster.

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Then you would find only another rational root #x=-1/3#