# How do you find the quotient of (x^3+27) / (x+3)?

Nov 9, 2015

${x}^{2} - 3 x + 9$

#### Explanation:

You can use long division of polynomials. To start set up the problem in long division format, with the denominator out in front and the numerator under the division sign.

$\textcolor{w h i t e}{\frac{x}{\textcolor{b l a c k}{x + 3}}} \frac{}{\text{)} {x}^{3} + 0 {x}^{2} + 0 x + 27}$

To start the division, look at the first term of each term. What do you need to multiply $x$ by to get ${x}^{3}$? The answer is ${x}^{2}$, so that goes on the top.

$\textcolor{w h i t e}{\frac{x}{\textcolor{b l a c k}{x + 3}}} \frac{{x}^{2} \textcolor{w h i t e}{- 3 x - 9 + 27 x}}{\text{)} {x}^{3} + 0 {x}^{2} + 0 x + 27}$

Now multiply the divisor, $x + 3$, by ${x}^{2}$ and subtract from the dividend. Remember, subtraction happens in columns.

$\textcolor{w h i t e}{\frac{x}{\textcolor{b l a c k}{x + 3}}} \frac{{x}^{2} \textcolor{w h i t e}{- 3 x - 9 + 27 x}}{\text{)} {x}^{3} + 0 {x}^{2} + 0 x + 27}$
$\textcolor{w h i t e}{X X X x} \frac{{x}^{3} + 3 {x}^{2}}{\textcolor{w h i t e}{{x}^{3}} - 3 {x}^{2}}$

Now we need to know what to multiply $x$ by to get $- 3 {x}^{2}$. Its $- 3 x$. Write $- 3 x$ on top, multiply by $x + 3$ and subtract.

$\textcolor{w h i t e}{\frac{x}{\textcolor{b l a c k}{x + 3}}} \frac{{x}^{2} - 3 x \textcolor{w h i t e}{+ 9 + 27 x}}{\text{)} {x}^{3} + 0 {x}^{2} + 0 x + 27}$
$\textcolor{w h i t e}{X X X x} \frac{{x}^{3} + 3 {x}^{2}}{\textcolor{w h i t e}{{x}^{3}} - 3 {x}^{2}}$
color(white)(XXXXX|)(-3x^2-9x)/(color(white)(-3x^2-)9x

Last one, we multiply $x$ by $9$ to get $9 x$.

$\textcolor{w h i t e}{\frac{x}{\textcolor{b l a c k}{x + 3}}} \frac{{x}^{2} - 3 x + 9 \textcolor{w h i t e}{+ 27 x}}{\text{)} {x}^{3} + 0 {x}^{2} + 0 x + 27}$
$\textcolor{w h i t e}{X X X x} \frac{{x}^{3} + 3 {x}^{2}}{\textcolor{w h i t e}{{x}^{3}} - 3 {x}^{2}}$
color(white)(XXXXX|)(-3x^2-9x)/(color(white)(-3x^2-)9x
$\textcolor{w h i t e}{X X X X X X X X X x} \frac{9 x + 27}{\textcolor{w h i t e}{9 x +} 0}$

There is no remainder, so the quotient is;

${x}^{2} - 3 x + 9$