How do you use synthetic division to divide #x^3 + 2x^2 - 2x + 24# by #x+4#?

2 Answers

Answer:

#frac{x^3 + 2x^2 - 2x + 24}{x + 4} = x^2 - 2x + 6#

Explanation:

Divide #x^3# by #x + 4#
Quotient #= x^2#
Remainder #= x^3 + 2x^2 - 2x + 24 - x^3 - 4 x ^2#
Remainder #= - 2x^2 - 2x + 24#

Divide #-2x^2# by #x + 4#
Quotient #= -2x#
Remainder #= - 2x^2 - 2x + 24 + 2x^2 + 8x#
Remainder #= 6x + 24#

Divide #6x# by #x + 4#
Quotient #=6#
Remainder #= 0#

May 18, 2018

Answer:

Synthetic division gives: #x^2-2x+6#

Process shown in detail

Explanation:

Given: #(x^3+2x^2-2x+24)/(x+4)#

Consider the denominator #x+4#

Set #x+4=0 =>color(red)(x=-4)#

Now we construct the coefficient manipulation.

#color(white)("dddd") x^3+2x^2-2x+24#
#color(white)("ddd")darrcolor(white)("d.d") darrcolor(white)("ddd")darrcolor(white)("ddd")darr#

#color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
bring down the first value of #color(magenta)(1)#

#color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24#
#color(white)("d..d")ul(|color(magenta)(darr )#
#color(white)("dddd")color(magenta)(1)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#-4xxcolor(magenta)(1)=color(blue)(-4)#
Place the #color(blue)(-4)# under the 2

#color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24#
#color(white)("d..d")ul(|color(magenta)(darr )color(white)("d")color(blue)(-4) )#
#color(white)("dddd")color(magenta)(1)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Add the 2 and the #color(blue)(-4)#

#color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24#
#color(white)("d..d")ul(|color(magenta)(darr )color(white)("d")color(blue)(-4) )#
#color(white)("dddd")color(magenta)(1)color(white)("d")-2#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(red)(-4)xx" the new "-2=+8#
Place the #-8# under the #-2#

#color(red)(-4)|bar(1color(white)("dddd")2color(white)("ddd")-2color(white)("ddd")24#
#color(white)("d..d")ul(|darr color(white)("d")-4color(white)("ddddd")8 )#
#color(white)("dddd")1color(white)("dd")-2#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Add the #-2" and the "8#

#color(red)(-4)|bar(1color(white)("dddd")2color(white)("ddd")-2color(white)("ddd")24#
#color(white)("d..d")ul(|darr color(white)("d")-4color(white)("ddddd")8 )#
#color(white)("dddd")1color(white)("dd")-2color(white)("ddddd")6#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(red)(-4)xx "the new "6 = -24#
Place the -24 under the existing +24
Add and you get the 0 so it is an exact division. No remainder.

#color(red)(-4)|bar(1color(white)("dddd")2color(white)("ddd")-2color(white)("ddd")24#
#color(white)("d..d")ul(|darr color(white)("d")-4color(white)("ddddd")8color(white)("ddd")24 )#
#color(white)("dddd")1color(white)("dd")-2color(white)("ddddd")6color(white)("ddd")0#
#color(white)("ddd.")darrcolor(white)("ddd")darrcolor(white)("dddd")darr #

#color(white)("dddd")x^2color(white)("d")-2xcolor(white)("dd")+6#