# How do you divide (3x^4+22x^3+ 15 x^2+26x+8)/(x-2) ?

Dec 28, 2015

Long divide the coefficients to find:

$\frac{3 {x}^{4} + 22 {x}^{3} + 15 {x}^{2} + 26 x + 8}{x - 2} = 3 {x}^{3} + 28 {x}^{2} + 71 x + 168$

with remainder $344$

#### Explanation:

There are other ways, but I like to long divide the coefficients like this:

A more compact form of this is called synthetic division, but I find it easier to read the long division layout.

Reassembling polynomials from the coefficients, we find:

$\frac{3 {x}^{4} + 22 {x}^{3} + 15 {x}^{2} + 26 x + 8}{x - 2} = 3 {x}^{3} + 28 {x}^{2} + 71 x + 168$

with remainder $344$.

Or you can write:

$3 {x}^{4} + 22 {x}^{3} + 15 {x}^{2} + 26 x + 8$

$= \left(x - 2\right) \left(3 {x}^{3} + 28 {x}^{2} + 71 x + 168\right) + 344$

Note that if you are long dividing polynomials that have a 'missing' term, then you need to include a $0$ for the coefficient of that term. We don't need to do that for our example.

As a check, let $f \left(x\right) = 3 {x}^{4} + 22 {x}^{3} + 15 {x}^{2} + 26 x + 8$ and evaluate $f \left(2\right)$, which should be the remainder:

$f \left(2\right) = 3 \cdot 16 + 22 \cdot 8 + 15 \cdot 4 + 26 \cdot 2 + 8$

$= 48 + 176 + 60 + 52 + 8$

$= 344$