How do you use the rational root theorem to list all possible rational roots?

1 Answer
George C.
Oct 25, 2015

See explanation...

Explanation:

Given a polynomial in the form:

#f(x) = a_0x^n + a_1x^(n-1) +...+ a_(n-1)x + a_n#

Any rational roots of #f(x) = 0# are expressible as #p/q# in lowest terms, where #p, q in ZZ#, #q != 0#, #p# a divisor of #a_n# and #q# a divisor of #a_0#.

To list the possible rational roots, identify all of the possible integer factors of #a_0# and #a_n#, and find all of the distinct fractions #p/q# that result.

For example, suppose #f(x) = 6x^3-12x^2+5x+10#

Then:

#a_n = 10# has factors #+-1#, #+-2#, #+-5# and #+-10#. (possible values of #p#)

#a_0 = 6# has factors #+-1#, #+-2#, #+-3# and #+-6#. (possible values of #q#)

Skip any combinations that have common factors (e.g. #10/6#) and list the resulting possible fractions:

#+-1/6#, #+-1/3#, #+-1/2#, #+-2/3#, #+-5/6#, #+-1#, #+-5/3#, #+-2#, #+-5/2#, #+-10/3#, #+-5#, #+-10#

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