# How do you divide (x^2 - 2x - 15)/(x + 3) using polynomial long division?

Dec 18, 2015

$x - 5$

Have a look at the method. It shows a useful 'trick'.

#### Explanation:

Given: $\frac{{x}^{2} - 2 x - 15}{x + 3} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left(1\right)$

Not all questions permit this approach of solution!

Consider $\textcolor{w h i t e}{. .} {x}^{2} - 2 x - 15$

This can be factored into:

$\left(x - 5\right) \left(x + 3\right) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left(2\right)$

Substitute expression (2) into expression (1)

$\frac{\left(x - 5\right) \left(x + 3\right)}{x + 3}$

Write as: $\frac{x + 3}{x + 3} \times \left(x - 5\right)$

But $\frac{x + 3}{x + 3}$ has the value of 1 giving

$1 \times \left(x - 5\right)$

$x - 5$

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$\textcolor{b l u e}{\text{Foot note}}$

Consider: $\frac{x + 3}{x + 3}$

If you were investigating values then this produces a problem.
You are not mathematically allowed to divide by 0.

So $\frac{x + 3}{x + 3}$ is 'Undefined' at $x = - 3$

For this very reason $\frac{0}{0}$ does $\underline{\textcolor{red}{\text{not equal 1}}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Jun 21, 2018

Using polynomial long division.

$x - 5$

#### Explanation:

Given: $\left({x}^{2} - 2 x - 15\right) \div \left(x + 3\right)$

$\textcolor{w h i t e}{\text{ddddddd.dd}} {x}^{2} - 2 x - 15$
$\textcolor{m a \ge n t a}{x} \left(x + 3\right) \to \underline{{x}^{2} + 3 x \leftarrow \text{ Subtract}}$
$\textcolor{w h i t e}{\text{dddddddddd") 0 color(white)(",}} - 5 x - 15$
$\textcolor{m a \ge n t a}{- 5} \left(x + 3\right) \to \textcolor{w h i t e}{\text{d") ul(-5x-15 larr" Subtract}}$
$\textcolor{w h i t e}{\text{dddddddddddddd}} 0 + 0$

$\left({x}^{2} - 2 x - 15\right) \div \left(x + 3\right) = \textcolor{m a \ge n t a}{x - 5}$