What is #(8x^3-4x^2-14x-20)/(2x-4)#?

2 Answers
Mar 22, 2018

#4x^2+6x+5#

Explanation:

#(8x^3-4x^2-14x-20)/(2x-4)#

#=(8x^3-16x^2+12x^2-24x+10x-20)/[(2)(x-2)]#

#=[(2)(4x^2+6x+5)(x-2)]/[(2)(x-2)]#

#=4x^2+6x+5#

Mar 22, 2018

#2x^3+6x^2+5#

Explanation:

#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#
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OR YOU CAN DO POLYNOMIAL LONG DIVISION