#color(blue)("Looking for possible cancelling out")#
At some point the cubic will cross the x-axis
Lets see if we can factor the cubic.#" "8x^3-4x^2-14x-20=0#
There are various factors of 20 including:
#1xx20#
#2xx10#
If we substitute #x+-1# it does not work
Lets try #x=+2#
#8(8)-4(4)-14(2)-20#
#64-16-28-20= 0 color(red)(larr" so "x=2" is a slution"#
A factor is #(x-2)(?......)#
We need #8x^3# as the first term
#->(color(red)(x)-2)(color(red)(8x^2)+ ....?)=color(red)(8x^3)-16x^2+...?#
We need to change the #-16x^2# into #-4x^2#. So to bring this about we need to generate the correction #+12x^2#. So the next stage of the build is:
#(color(red)(x)-2)(8x^2color(red)(+12x)+ ....?)=8x^3-ubrace(16x^2color(red)(+12x^2))-24x+..?#
#(x-2)(8x^2+12x+ ....?)=8x^3color(white)("ddd") -4x^2color(white)("ddd")-24x+..?#
We need to change the #-24x# into #-14x# so we need to generate #+10x#
#(color(red)(x)-2)(8x^2+12xcolor(red)(+10))= 8x^3-4x^2-ubrace(24xcolor(red)(+10x))-20#
#(x-2)(8x^2+12x+10)= 8x^3-4x^2color(white)("dd")-14xcolor(white)("dd")-20#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Putting it all together")#
#((x-2)(8x^2+12x+10))/(2x-4)#
Factor out the 2 in the denominator
#((x-2)(8x^2+12x+10))/(2(x-2)) color(white)("dd") ->color(white)("dd") (cancel((x-2))(8x^2+12x+10))/(2cancel((x-2))#
#color(white)("dddddddddddddddddddddd")->color(white)("dddddddd")2x^3+6x^2+5#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
OR YOU CAN DO POLYNOMIAL LONG DIVISION
