How do you use the rational root theorem to find the roots of #x^3-9x^2+14x=48#?

1 Answer
Feb 3, 2016

Answer:

Test all possible values of #x in {+- ("factors of " 48)/("factors of "1)}#
Any that satisfy the equation are roots of the equation.
#color(white)("XXX")#(The only Real root is at #x=8# in this case).

Explanation:

The candidates for rational roots of #color(red)(1)x^3-9x^2+14x-color(blue)(48)=0#
(notice, I've slightly re-arranged the equation)
are #+-("factors of "color(blue)(48))/("factors of "color(red)(1))#
#color(white)("XXXXXXXXXXXX")= +- (1,2,3,4,6,8,12,16,24,48)#

Therefore there are #20# candidates to test!
(It is useful if you can set this up as a spreadsheet or computer program.)

Here are the evaluations (using synthetic substitution/division) for the first #8# candidates:
enter image source here
This gives us #x=8# as a zero of the given equation.

Not of the subsequent factors gave a result of #0# (Take my word for it).

It might be noted that treating these entries as synthetic division gives:
#color(white)("XXX")x^3-9x^2+14x-48 = (x-8)(x^2-x+6)#
and taking the discriminant of #(x^2-x+6)# shows us that there are no further Real roots (i.e. no other Real zeros)