How do you list all possible rational roots for each equation, use synthetic division to find the actual rational root, then find the remaining 2 roots for #2x^3-7x^2-46x-21=0#?

1 Answer
Feb 3, 2016

Answer:

Assume that one of the rational roots is integer;
then perform synth.div for each integer root of #21#
Final answer: roots #in {-3,7,-1/2}#

Explanation:

The integer factors of #21# are #{1,3,7,-1,-3,-7}#
Performing the synthetic division with #(x+f)#
for each #f# which is an integer factor of #21#:
enter image source here
We notice that synthetic division of #color(green)(2x^3-7x^2-46x-12)#
by #(x+3)# gives #color(blue)(2x^2-13x-7)# with a Remainder of #color(red)(0)#

So #color(green)(2x^3-7x^2-46x-12)=(x+3)(color(blue)(2x^2-13x-7))#

We can then factor #((color(blue)(2x^2-13x-7))# as
#color(white)("XXX")(2x+1)(x-7)#

So #color(green)(2x^3-7x^2-46x-12)=(x+3)(2x+1)(x-7)#
and
the roots are:
#color(white)("XXX"){-3,-1/2,+7}#
(the roots are the values of #x# that make the factors #=0#)