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)(6x5−2x3+4x2−3x+1)÷(x−2)?

1 Answer
Alan P.
Oct 23, 2015

Remainder =187=187

Explanation:

The Remainder Theorem says that the remainder of a polynomial f(x)f(x) by (x-a)(x−a) is f(a)f(a)
and one way we can evaluate f(a)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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