A polynomial is a finite expression consisting of the sum of products between constants and variables which have positive whole number indices. For instance, $5x^2 -3x+7$ is a polynomial, while $\sqrt{x}$ is not a polynomial. You can divide polynomial by other polynomials, in particular, by linear polynomials.

Division of polynomials using long division

Procedure

1.	Identify the numerator and the denominator in your division.
2.	Write down the elements of your division: write the numerator inside the "bus stop" and write the denominator outside it to the left.
3.	Compare the first element of the numerator and denominator, to find out the multiplier (the term you multiply the first element of the denominator by to get the first element of the numerator). Write this above the "bus stop" in-line with the like-term.
4.	Multiply the whole denominator by the multiplier found in step 3 and write this beneath the numerator such that like-terms are in-line. Subtract that from the numerator and write the result beneath, again keeping like-terms in-line.
5.	Repeat steps from 1-4 until you reach the remainder. Then, write your answer in this form: $\cfrac{\rm numerator}{\rm denominator} = \rm quotient + \cfrac{\rm remainder}{\rm denominator}$

Example 1

Divide $x^3 +x + 10$ by $x+2$ .

First, identify and write down the elements of your division. It is useful to write each term, even if the coefficient is zero:

$\begin{array}{r}x+2 {\overline{\smash{\big)}\, x^3+ 0x^2 + x +10}}\end{array}$

Compare the first elements of the numerator and the denominator, and write down the multiplier that takes this denominator element ( $x$ ) to the numerator element ( $x^3$ ). Write this multiplier ( $x^2$ ) above the "bus stop".

$\begin{array}{r}x^2 \hspace{14mm} \\x+2 {\overline{\smash{\big)}\,x^3+0x^2+x+10}}\end{array}$

Multiply the denominator by the multiplier found in the previous step, writing it beneath the latest line in the "bus stop" and subtract the result from the this latest line.

$\begin{array}{r}x^2 \hspace{14mm} \\x+2 {\overline{\smash{\big)}\, x^3 + 0x^2 + x + 10}}\\\hspace{0mm} x^3 + 2x^2\hspace{14mm}\\\overline{\smash{}\, -2x^2 +x +10}\end{array}$

Repeat these steps until you reach the final answer.

$\begin{array}{r}x^2 -2x +5 \hspace{0mm} \\x+2 {\overline{\smash{\big)}\, x^3 + 0x^2 + x + 10}}\\ \hspace{0mm} x^3 + 2x^2\hspace{14mm}\\\overline{\smash{}\, \hspace{4mm}-2x^2 +x +10}\\\hspace{0mm} 2x^2 - 4x \hspace{6mm}\\\overline{\smash{}\,\hspace{12mm} 5x +10}\\\hspace{0mm} 5x+10 \\\overline{\smash{}\, \hspace{9.5mm}0}\\\end{array}$

Now, the term above the "bus stop" is the result. Express the result in the simplified form.

$\underline{\cfrac{x^3 + x + 10}{x+2}= x^2 - 2x + 5}$ 

Example 2

Divide $6x^5 + 13x^4 -4x^3 -9x^2 +21x +19$ by $2x+3$ .

First, identify and write down the elements of your division.

 $\begin{array}{r}2x+3 {\overline{\smash{\big)}\, 6x^5 + 13 x^4 - 4x^3 -9x^2 +21x +19}}\end{array}$ 

Compare the first elements of the numerator and the denominator, and write down the multiplier.

 $\begin{array}{r}3x^4 \hspace{37mm}\\ 2x+3 {\overline{\smash{\big)}\, 6x^5 + 13 x^4 - 4x^3 -9x^2 +21x +19}}\end{array}$ 

Multiply the denominator by the quantity found in the previous step, and subtract the result from the line above.

$\begin{array}{r}3x^4 \hspace{37mm}\\ 2x+3 {\overline{\smash{\big)}\, 6x^5 + 13 x^4 - 4x^3 -9x^2 +21x +19}}\\\hspace{55mm}6x^5 + 9x^4 \hspace{38.5mm}\\\overline{\smash{}\, 4x^4 - 4x^3 -9x^2 +21 x + 19 \hspace{2mm}}\\\end{array}$

Repeat these steps until you reach the final answer.

$\begin{array}{r}3x^4 + 2x^3 -5x^2 +3x + 6 \hspace{2mm}\\ 2x+3 {\overline{\smash{\big)}\, 6x^5 + 13 x^4 - 4x^3 -9x^2 +21x +19}}\\ \hspace{55mm}6x^5 + 9x^4 \hspace{38.5mm}\\\overline{\smash{}\, \hspace{11mm}4x^4 - 4x^3 -9x^2 +21 x + 19 \hspace{0mm}}\\ \hspace{47mm}4x^4+6x^3 \hspace{27mm}\\\overline{\smash{}\, \hspace{3mm}- 10x^3 -9x^2 +21 x + 19 \hspace{0mm}}\\ \hspace{40mm}-10x^3 - 15x^2 \hspace{16mm}\\\overline{\smash{}\, \hspace{14mm}6x^2 +21 x + 19 \hspace{0mm}}\\ \hspace{20mm}6x^2 + 9x\hspace{9.5mm}\\\overline{\smash{}\, \hspace{9mm}12 x + 19}\\\hspace{11mm}12x + 18 \\\overline{\smash{}\, \hspace{11mm}1\hspace{0mm}}\\\end{array}$

Now, express the result in the divided form with the remainder.

$\underline{\cfrac{6x^5 + 13x^4 - 4x^3 - 9x^2 + 21x + 19}{2x+3} = 3x^4 + 2x^3 - 5x^2 +3x +6 + \cfrac{1}{2x+3}}$

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