# Long Division of Polynomials

## Key Questions

• You can use the Factor Theorem or Synthetic Division on polynomials to find the quotient using long division.

Please see the tutorial for step-by-step instructions.

Here are a couple of examples...

#### Explanation:

Here's a sample animation of long dividing ${x}^{3} + {x}^{2} - x - 1$ by $x - 1$ (which divides exactly).

Write the dividend under the bar and the divisor to the left. Each is written in descending order of powers of $x$. If any power of $x$ is missing, then include it with a $0$ coefficient. For example, if you were dividing by ${x}^{2} - 1$, then you would express the divisor as ${x}^{2} + 0 x - 1$.

Choose the first term of the quotient to cause leading terms to match. In our example, we choose ${x}^{2}$, since $\left(x - 1\right) \cdot {x}^{2} = {x}^{3} - {x}^{2}$ matches the leading ${x}^{3}$ term of the dividend.

Write the product of this term and the divisor below the dividend and subtract to give a remainder ($2 {x}^{2}$).

Bring down the next term ($- x$) from the divisor alongside it.

Choose the next term ($2 x$) of the quotient to match the leading term of this remainder, etc.

Stop when there is nothing more to bring down from the dividend and the running remainder has lower degree than the divisor.

In our example, the division is exact. We are left with no remainder.

Instead of writing out all of the terms in full, you can just write out and divide the coefficients. For example:

Here we divide $3 {x}^{4} + 2 {x}^{3} - 11 {x}^{2} - 2 x + 5$ by ${x}^{2} - 2$ to get $3 {x}^{2} + 2 x - 5$ with remainder $2 x - 5$.

#### Explanation:

Given: What is long division of polynomials?

Long division of polynomials is very similar to regular long division. It can be used to simplify a rational function $\frac{N \left(x\right)}{D \left(x\right)}$ for integration in Calculus, to find a slant asymptote in PreCalculus, and many other applications. It is done when the denominator polynomial function has a lower degree than the numerator polynomial function. The denominator can be a quadratic.

Ex. $y = \frac{{x}^{2} + 12}{x - 2}$

" "ul(" "x + 2" ")
$x - 2 | {x}^{2} + 0 x + 12$
$\text{ } \underline{{x}^{2} - 2 x}$
$\text{ } 2 x + 12$
" "ul(2x -4" ")
$\text{ } 16$

This means $y = \frac{{x}^{2} + 12}{x - 2} = x + 2 + \frac{16}{x - 2}$

The slant asymptote in the above example is $y = x + 2$