How do you use synthetic division to divide #(x^3 -8 x^2 - 25x + 203)# by #x-5#?

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Answer 1 · 2015-05-23T00:36:02

Synthetic division is somewhat like long division.

Starting with #(x^3-8x^2-25x+203)#, first look for a multiplier for #(x-5)# that will cause give a match for the highest order term.

Choose #x^2# as the first multiplier.

#x^2(x-5) = x^3-5x^2#

Subtract this from our original polynomial to get the remainder:

#(x^3-8x^2-25x+203) - (x^3 - 5x^2)#

#= (-3x^2-25x+203)#

Now choose a multiplier #(-3x)# for #(x-5)# to match the leading term #-3x^2# of the remainder...

#(-3x)(x-5) = (-3x^2+15x)#

Subtract this from our remainder to get a new remainder:

#(-3x^2-25x+203) - (-3x^2+15x) = (-40x+203)#

Now choose a multiplier #(-40)# for #(x-5)# to match the leading term #-40x# of our remainder...

#(-40)(x - 5) = (-40x+200)#

Subtract this from our remainder to get a new remainder:

#(-40x+203)-(-40x+200) = 3#

Adding our multipliers together, we find:

#x^3-8x^2-25x+203 = (x-5)(x^2-3x-40) + 3#