How do you divide (x^3+x^2+3x+1)/(x-3)?

Jul 14, 2017

${x}^{2} + 4 x + 15 \text{ rem } \frac{46}{x - 3}$

Explanation:

Division by synthetic division is the simplest and quickest method in this case. We will only use the numerical coefficients.

$x - 3 = 0 \text{ "rarr x =3" }$ goes outside on the left.

$\frac{1 {x}^{3} + 1 {x}^{2} + 3 x + 1}{x - 3}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\text{ "3|" "1" "1" "3" } 1$
$\text{ "|ul" "darrul" }$
$\text{ } 1 \textcolor{w h i t e}{\times \times \times \times \times \times \times x} \leftarrow$ bring down the $1$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\text{ "color(red)(3)|" "1" "1" "3" } 1$
$\text{ "|ul" "darrul" } \textcolor{b l u e}{3} \underline{\textcolor{w h i t e}{\times \times \times \times \times x}} \leftarrow \textcolor{red}{3 \times 1} = \textcolor{b l u e}{3}$
$\text{ "color(red)(1)" } \textcolor{b l u e}{4} \textcolor{w h i t e}{\times \times \times \times \times x} \leftarrow 1 + \textcolor{b l u e}{3 = 4}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\text{ "color(red)(3)|" "1" "1" "3" } 1$
" "|ul" "darrul" "3ul" "color(green)(12)ulcolor(white)(xxxxxxxxx)larrcolor(red)(3xxcolor(blue)(4) color(green)(=12)
$\text{ "1" "color(blue)(4)" } \textcolor{g r e e n}{15} \textcolor{w h i t e}{\times \times \times \times x} \leftarrow 3 + \textcolor{g r e e n}{12 = 15}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\text{ "color(red)(3)|" "1" "1" "3" } 1$
" "|ul" "darrul" "3ul" "12" "color(magenta)(45)ulcolor(white)(xxxxxx)larrcolor(red)(3xxcolor(green)(15) color(magenta)(=45)
$\text{ "1" "color(blue)(4)" "color(green)(15)" } \textcolor{m a \ge n t a}{46} \textcolor{w h i t e}{\times \times \times} \leftarrow 1 + \textcolor{m a \ge n t a}{45 = 46}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times} \uparrow$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times x} \text{remainder}$

The bottom line gives the numerical coefficients of the quotient.
${x}^{3} \div x = {x}^{2}$

Quotient:

$1 {x}^{2} + \textcolor{b l u e}{4} x + \textcolor{g r e e n}{15} \text{ rem } \textcolor{m a \ge n t a}{46}$

Jul 14, 2017

The remainder is $46$ and the quotient is $= {x}^{2} + 4 x + 15$

Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a a a}$$3$$\textcolor{w h i t e}{a a a a a a a}$$1$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$_________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a}$$3$$\textcolor{w h i t e}{a a a a a}$$12$$\textcolor{w h i t e}{a a a a a a}$$45$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a}$$1$$\textcolor{w h i t e}{a a a}$$4$$\textcolor{w h i t e}{a a a a a}$$15$$\textcolor{w h i t e}{a a a a a a}$$\textcolor{red}{46}$

The remainder is $46$ and the quotient is $= {x}^{2} + 4 x + 15$

$\frac{{x}^{3} + 3 {x}^{2} + 3 x + 1}{x - 3} = {x}^{2} + 4 x + 15 + \frac{46}{x - 3}$