# 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)?

Oct 23, 2015

Remainder $= 187$

#### Explanation:

The Remainder Theorem says that the remainder of a polynomial $f \left(x\right)$ by $\left(x - a\right)$ is $f \left(a\right)$
and one way we can evaluate $f \left(a\right)$ 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.