How do you use the rational roots theorem to find all possible zeros of #f(x)=x^3+10x^2-13x-22 #?

1 Answer
George C.
Jun 5, 2016

#x = -1#, #x=2# and #x=-11#

Explanation:

#f(x) = x^3+10x^2-13x-22#

By the rational root theorem, any rational zeros of #f(x)# must be expressible in the form #p/q# for integers #p, q# with #p# a divisor of the constant term #-22# and #q# a divisor of the coefficient #1# of the leading term.

That means that the only possible rational zeros are:

#+-1#, #+-2#, #+-11#, #+-22#

Trying each in turn, we find:

#f(-1) = -1+10+13-22 = 0#

#f(2) = 8+40-26-22 = 0#

#f(-11) = -1331+1210+143-22 = 0#

So we have found all of the zeros:

#x = -1#, #x=2# and #x=-11#

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