How do you use the remainder theorem and synthetic division to find the remainder when #(6x^5 - 2x^3 + 4x^2 - 3x + 1) div (x - 2)#?

1 Answer
Alan P.
Oct 23, 2015

Remainder #=187#

Explanation:

The Remainder Theorem says that the remainder of a polynomial #f(x)# by #(x-a)# is #f(a)#
and one way we can evaluate #f(a)# using "synthetic substitution"

"Synthetic division" is an alternate (but identical) method of combining the Remainder Theorem and "Synthetic substitution"

Here is what it looks like for the given example

#{: (,,x^5,x^4,x^3,x^2,x^1,x^0), (,"|",6,0,-2,+4,-3,+1), (+2,"|",,12,24,44,96,186), ("-----",,"-----","-----","-----","-----","-----","-----"), (,,6,12,22,48,93,), (,,x^4,x^3,x^2,x^1,x^0,R=197) :}#

Note: the powers of #x# are normally not actually written; I've simply included them to (hopefully) improve the clarity.

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