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## Synthetic Division

We just finished talking about synthetic division in the College Algebra course today and got into a discussion of how to represent the remainder.  For example, given the problem:

$\frac{x^4-2x^3-x+10}{x-2}$

The answer turns out to be: $x^3-1 R: 8$ or you can say that the answer is: $x^3-1+\frac{8}{x-2}$.  This all goes back to the division algorithm which says that given two numbers $a$ and $b$, then solving the problem $\frac{a}{b}$ means finding $q$, the quotient and $r$, the remainder such that $a=b*q+r$ (with $r) .

If we take the expression $a=b*q+r$ and divide on both sides by $b$, then we’ll have $\frac{a}{b}=\frac{b*q}{b}+\frac{r}{b}$ or $\frac{a}{b}=q+\frac{r}{b}$.

Which of these forms we prefer depends on whether we want to say that:

$x^4-2x^3-x+10=(x-2)(x^3-1)+8$

or

$\frac{x^4-2x^3-x+10}{x-2}=x^3-1+\frac{8}{x-2}$

### One Response

1. thank for sharing this. Really helpul info on synthetic division