How do you long divide $\frac{x^{2} + 4 x + 4}{x + 2}$ $(x^2+4x+4) / (x+2)$ ?

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How do you long divide $\frac{x^{2} + 4 x + 4}{x + 2}$ $(x^2+4x+4) / (x+2)$ ?

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Answer 1 · 2015-07-16T15:09:59

It is a challenge to format and there are various ways of writing the details. Here's the one that I learned and teach (using a standard textbook in the US):

Explanation:

To divide $\frac{x^{2} + 4 x + 4}{x + 2}$ $(x^2 +4x + 4)/(x+2)$

Write the divisor, $x + 2$ $x+2$ on the left. Then the dividend, $x^{2} + 4 x + 4$ $x^2+4x+4$ under the division thing. (I don't know its name.)

$" " " "$ $- - - - -$ $-----$
$x + 2)$ $x+2 )$ $x^{2}$ $x^2$ $+ 4 x$ $+4x$ $+ 4$ $+4$

Look at the first terms on the divisor $x + 2$ $x+2$ and the dividend $x^{2} + 4 x + 4$ $x^2+4x+4$ . What do we need to multiply $x$ $x$ by, to get $x^{2}$ $x^2$ ? We need to multiply by $x$ $x$ . Write that $x$ $x$ on the top line:

$x$ $" " " " " " "x$
$" " " "$ $- - - - -$ $-----$
$x + 2)$ $x+2 )$ $x^{2}$ $x^2$ $+ 4 x$ $+4x$ $+ 4$ $+4$

Now multiply $x$ $x$ times the divisor, $x + 2$ $x+2$ , to get $x^{2} + 2 x$ $x^2+2x$ and write that under the dividend.

$" " " " " " "$ $x$ $x$
$" " " "$ $- - - - -$ $-----$
$x + 2)$ $x+2 )$ $x^{2}$ $x^2$ $" "$ $+ 4 x$ $+4x$ $" "$ $+ 4$ $+4$
$" " " " "" "$ $x^{2}$ $x^2$ $" "$ $+ 2 x$ $+2x$
$" " " "$ $- - - - -$ $-----$

Now we need to subtract $x^{2} + 2 x$ $x^2+2x$ from the dividend. (You may find it simpler to change the signs and add.)

$" " " " " " "$ $x$ $x$
$" " " "$ $- - - - -$ $-----$
$x + 2)$ $x+2 )$ $x^{2}$ $x^2$ $" "$ $+ 4 x$ $+4x$ $" "$ $+ 4$ $+4$
$" " " "$ $- x^{2} - 2 x$ $color(red)(-)x^2color(red)(-)2x$
$" " " "$ $- - - - -$ $-----$
$" "" "" "" "" "$ $2 x$ $2x$ $" "$ $+ 4$ $+4$

Now, look at the first terms of the divisor and the expression on the last line. What do we need to multiply $x$ $x$ (the first term of the divisor) by to get $2 x$ $2x$ (the first term on the last line)? We need to multiply by $2$ $2$
Write $+ 2$ $+2$ on the top line, then multiply $2$ $2$ times the divisor $x + 2$ $x+2$ , to get $2 x + 4$ $2x+4$ and write it underneath.

$" " " " " " "$ $x$ $x$ $" "$ $+ 2$ $+2$
$" " " "$ $- - - - -$ $-----$
$x + 2)$ $x+2 )$ $x^{2}$ $x^2$ $" "$ $+ 4 x$ $+4x$ $" "$ $+ 4$ $+4$
$" " " "$ $- x^{2} - 2 x$ $color(red)(-)x^2color(red)(-)2x$
$" " " "$ $- - - - - -$ $------$
$" " " " " "" "$ $2 x$ $2x$ $" "$ $+ 4$ $+4$
$" " " " " " " "$ $2 x$ $2x$ $" "$ $+ 4$ $+4$
$" " " "$ $- - - - - -$ $------$

Now subtract (change the signs and add), to get:

$" " " " " " "$ $x$ $x$ $" "$ $+ 2$ $+2$
$" " " "$ $- - - - -$ $-----$
$x + 2)$ $x+2 )$ $x^{2}$ $x^2$ $" "$ $+ 4 x$ $+4x$ $" "$ $+ 4$ $+4$
$" " " "$ $- x^{2} - 2 x$ $color(red)(-)x^2color(red)(-)2x$
$" " " "$ $- - - - - -$ $------$
$" " " " " "" "$ $2 x$ $2x$ $" "$ $+ 4$ $+4$
$" " " " " "$ $- 2 x$ $color(red)(-)2x$ $" "$ $- 4$ $color(red)(-)4$
$" " " "$ $- - - - - -$ $------$
$" " " " " " " " " "" " " "$ $0$ $0$

The quotient is $x + 2$ $x+2$ and the remainder is $0$ $0$ .

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Explanation:

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