How do you solve #x^3-3x^2-9x+27<0# using a sign chart?

1 Answer
Nov 12, 2016

Answer:

The answer is #x in ] -oo,-3 [#

Explanation:

Let #f(x)=x^3-3x^2-9x+27# and #f(x)<0#

Then, #f(-3)=-27-27+27+27=0#

So, #(x+3)# is a factor of #f(x)#

Let's do a long division

#color(white)(aaaa)##x^3-3x^2-9x+27##color(white)(aaaa)##∣##x+3#
#color(white)(aaaa)##x^3+3x^2##color(white)(aaaaaaaaaaaaa)##∣##x^2-6x+9#
#color(white)(aaaaa)##0-6x^2-9x#
#color(white)(aaaaaaa)##-6x^2-18x#
#color(white)(aaaaaaaaaaa)##0+9x+27#
#color(white)(aaaaaaaaaaaaa)##+9x+27#

#x^2-6x+9=(x-3)^2#

So, #f(x)=(x+3)(x-3)^2#

let's do the sign chart

#color(white)(aaaaa)##x##color(white)(aaaaa)##-oo##color(white)(aaaaa)##-3##color(white)(aaaaa)##3##color(white)(aaaaa)##+oo#
#color(white)(aaaaa)##x+3##color(white)(aaaaaaa)##-##color(white)(aaaaa)##+##color(white)(aaaaa)##+#
#color(white)(aaaaa)##(x-3)^2##color(white)(aaaaa)##+##color(white)(aaaaa)##+##color(white)(aaaaa)##+#
#color(white)(aaaaa)##f(x)##color(white)(aaaaaaaa)##-##color(white)(aaaaa)##+##color(white)(aaaaa)##+#

#f(x)<0# when #x in ] -oo,-3 [#

graph{x^3-3x^2-9x+27 [-76.4, 55.3, -32.1, 33.74]}