# How do you use long division to divide (x^4-2x^3-4x^2+2x+3) -: (x^2+2x+1)?

Sep 9, 2017

$\left({x}^{4} - 2 {x}^{3} - 4 {x}^{2} + 2 x + 3\right) \div \left({x}^{2} + 2 x + 1\right) = {x}^{2} - 4 x + 3$, with no remainder.

#### Explanation:

Set up the long division like this:

$\textcolor{m a \ge n t a}{{x}^{2}} + 2 x + 1 \overline{| \text{ } \textcolor{m a \ge n t a}{{x}^{4}} - 2 {x}^{3} - 4 {x}^{2} + 2 x + 3}$

Divide $\textcolor{m a \ge n t a}{{x}^{4} / {x}^{2}}$, giving $\textcolor{red}{{x}^{2}}$; put this quotient above the ${x}^{4}$:

color(white)(x^2+2x+1bar(|"  "color(red)(x^2)
${x}^{2} + 2 x + 1 \overline{| \text{ } {x}^{4} - 2 {x}^{3} - 4 {x}^{2} + 2 x + 3}$

Multiply $\textcolor{red}{{x}^{2}} \times \left({x}^{2} + 2 x + 1\right)$, giving $\textcolor{b l u e}{{x}^{4} + 2 {x}^{3} + {x}^{2}}$; put this product below the ${x}^{4} - 2 {x}^{3} - 4 {x}^{2}$:

color(white)(x^2+2x+1bar(|"  "color(black)(x^2)
${x}^{2} + 2 x + 1 \overline{| \text{ } {x}^{4} - 2 {x}^{3} - 4 {x}^{2} + 2 x + 3}$
color(white)(x^2+2x+1bar(|"  "color(blue)(x^4+2x^3+color(white)(1)x^2))

Subtract $\left({x}^{4} - 2 {x}^{3} - 4 {x}^{2}\right) - \left(\textcolor{b l u e}{{x}^{4} + 2 {x}^{3} + {x}^{2}}\right)$, giving color(orange)(–4x^3-5x^2); draw a line under $\textcolor{b l u e}{{x}^{4} + 2 {x}^{3} + {x}^{2}}$ and write this difference below the line:

color(white)(x^2+2x+1bar(|"  "color(black)(x^2)
${x}^{2} + 2 x + 1 \overline{| \text{ } {x}^{4} - 2 {x}^{3} - 4 {x}^{2} + 2 x + 3}$
color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 |} \overline{\text{ "color(white)(x^4" ")color(orange)(-4x^3-5x^2)" }}$

Copy the $\textcolor{g r e e n}{2 x}$ from the dividend down below this line:

color(white)(x^2+2x+1bar(|"  "color(black)(x^2)
${x}^{2} + 2 x + 1 \overline{| \text{ } {x}^{4} - 2 {x}^{3} - 4 {x}^{2} + \textcolor{g r e e n}{2 x} + 3}$
color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 |} \overline{\text{ "color(white)(x^4)-4x^3-5x^2" } \textcolor{g r e e n}{+ 2 x}}$

Repeat this process twice, dividing the latest leading term below your line by the leading ${x}^{2}$ from the divisor:

color(white)(x^2+2x+1bar(|"  "color(black)(x^2-color(red)(4x))
$\textcolor{m a \ge n t a}{{x}^{2}} + 2 x + 1 \overline{| \text{ } {x}^{4} - 2 {x}^{3} - 4 {x}^{2} + 2 x + 3}$
color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 |} \overline{\text{ } \textcolor{w h i t e}{{x}^{4}} - \textcolor{m a \ge n t a}{4 {x}^{3}} - 5 {x}^{2} + 2 x}$

...

color(white)(x^2+2x+1bar(|"  "color(black)(x^2-color(red)(4x))
$\textcolor{red}{{x}^{2} + 2 x + 1} \overline{| \text{ } {x}^{4} - 2 {x}^{3} - 4 {x}^{2} + 2 x + 3}$
color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 |} \overline{\text{ } \textcolor{w h i t e}{{x}^{4}} - 4 {x}^{3} - 5 {x}^{2} + 2 x}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \overline{\text{ } \textcolor{b l u e}{- 4 {x}^{3} - 8 {x}^{2} - 4 x}}}$

...

color(white)(x^2+2x+1bar(|"  "color(black)(x^2-4x)
${x}^{2} + 2 x + 1 \overline{| \text{ } {x}^{4} - 2 {x}^{3} - 4 {x}^{2} + 2 x + \textcolor{g r e e n}{3}}$
color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 |} \overline{\text{ } \textcolor{w h i t e}{{x}^{4}} - 4 {x}^{3} - 5 {x}^{2} + 2 x}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \overline{\text{ } \textcolor{b l a c k}{- 4 {x}^{3} - 8 {x}^{2} - 4 x}}}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \text{ }} \overline{\textcolor{w h i t e}{- 4 {x}^{3} +} \textcolor{\mathmr{and} a n \ge}{3 {x}^{2} + 6 x} + \textcolor{g r e e n}{3}}$

...

color(white)(x^2+2x+1bar(|"  "color(black)(x^2-4x"  "+color(red)3)
$\textcolor{m a \ge n t a}{{x}^{2}} + 2 x + 1 \overline{| \text{ } {x}^{4} - 2 {x}^{3} - 4 {x}^{2} + 2 x + 3}$
color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 |} \overline{\text{ } \textcolor{w h i t e}{{x}^{4}} - 4 {x}^{3} - 5 {x}^{2} + 2 x}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \overline{\text{ } \textcolor{b l a c k}{- 4 {x}^{3} - 8 {x}^{2} - 4 x}}}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \text{ }} \overline{\textcolor{w h i t e}{- 4 {x}^{3} +} \textcolor{m a \ge n t a}{3 {x}^{2}} + 6 x + 3}$

...

color(white)(x^2+2x+1bar(|"  "color(black)(x^2-4x"  "+color(red)(3))
$\textcolor{red}{{x}^{2} + 2 x + 1} \overline{| \text{ } {x}^{4} - 2 {x}^{3} - 4 {x}^{2} + 2 x + 3}$
color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 |} \overline{\text{ } \textcolor{w h i t e}{{x}^{4}} - 4 {x}^{3} - 5 {x}^{2} + 2 x}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \overline{\text{ } \textcolor{b l a c k}{- 4 {x}^{3} - 8 {x}^{2} - 4 x}}}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \text{ }} \overline{\textcolor{w h i t e}{- 4 {x}^{3} +} 3 {x}^{2} + 6 x + 3}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \text{ "bar(color(blue)(" } 3 {x}^{2} + 6 x + 3}$

...

color(white)(x^2+2x+1bar(|"  "color(black)(x^2-4x"  "+3)
${x}^{2} + 2 x + 1 \overline{| \text{ } {x}^{4} - 2 {x}^{3} - 4 {x}^{2} + 2 x + 3}$
color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 |} \overline{\text{ } \textcolor{w h i t e}{{x}^{4}} - 4 {x}^{3} - 5 {x}^{2} + 2 x}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \overline{\text{ } \textcolor{b l a c k}{- 4 {x}^{3} - 8 {x}^{2} - 4 x}}}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \text{ }} \overline{\textcolor{w h i t e}{- 4 {x}^{3} +} 3 {x}^{2} + 6 x + 3}$
$\textcolor{w h i t e}{{x}^{2} + 2 x + 1 | \text{ "bar(color(black)(" } 3 {x}^{2} + 6 x + 3}$
color(white)(x^2+2x+1|"                       ")bar(color(white)(3x^2+6x+color(orange)0)