Polynomial Long Division

Extending Integer Long Division to Polynomials

Polynomial long division finds the quotient and remainder by repeating the same divide, multiply, subtract, and bring-down pattern used in integer long division. Column alignment is what makes the pattern work, because only like terms may be subtracted from each other.

Preparing for Long Division

Before starting the division, there are a few things to prepare:

Order Terms:

Write the dividend ( $P(x)$ ) and the divisor ( $Q(x)$ ) in descending order of variable powers (from highest power to lowest).
Complete Terms:

If a power is missing, insert its zero coefficient as a placeholder. This keeps like powers in the same column during subtraction.

Example:

If $P(x) = 4x^4 + 17x^3 - 3x + 1$ , the $x^2$ term is missing.

So we write it as $P(x) = 4x^4 + 17x^3 + 0x^2 - 3x + 1$ .
Set Up Division:

Write the division in long division format, with $P(x)$ (the completed form) inside the division symbol and $Q(x)$ outside.

Steps for Long Division

The division process is performed step-by-step as follows:

Divide: Divide the first term of $P(x)$ by the first term of $Q(x)$ . Write the result as the first term of the quotient ( $H(x)$ ) above the line.
Multiply: Multiply the quotient term just obtained by the entire divisor $Q(x)$ .
Subtract: Write the result of the multiplication below $P(x)$ , aligning like terms, then subtract it from $P(x)$ to get a temporary remainder.
Bring Down: Bring down the next term from $P(x)$ next to the temporary remainder to form a new polynomial.
Repeat: Repeat steps $1\text{-}4$ with this new polynomial until the degree of the temporary remainder is less than the degree of the divisor $Q(x)$ .

Long Division Example

Divide $P(x) = 4x^4 + 17x^3 - 3x + 1$ by $Q(x) = x^2 + 4x - 1$ .

Preparation:
- $P(x) = 4x^4 + 17x^3 + 0x^2 - 3x + 1$ (complete the $x^2$ term)
- $Q(x) = x^2 + 4x - 1$
Division Process:

The quotient is $4x^2+x$ and the remainder is $-2x+1$ . The next steps show each multiplication and subtraction, including the zero coefficient of $x^2$ .
Step-by-Step Explanation:

Iteration $1$ :
- Divide: Divide the first term $4x^4$ by the first term of the divisor $x^2$ :
  
  $\frac{4x^4}{x^2} = 4x^2$
  
  Write $4x^2$ as the first term of the quotient.
- Multiply: Multiply $4x^2$ by the divisor $x^2 + 4x - 1$ :
  
  $\begin{aligned} &4x^2(x^2 + 4x - 1) \\ &= 4x^4 + 16x^3 - 4x^2 \end{aligned}$
- Subtract: Subtract the result from the initial polynomial:
  
  $\begin{aligned} &(4x^4 + 17x^3 + 0x^2) \\ &\quad{}- (4x^4 + 16x^3 - 4x^2) \\ &= x^3 + 4x^2 \end{aligned}$
- Bring Down: Bring down the next term ( $-3x$ ) to get the new polynomial:
  $x^3 + 4x^2 - 3x$
Iteration $2$ :
- Divide: Divide the first term of the new polynomial $x^3$ by the first term of the divisor $x^2$ :
  
  $\frac{x^3}{x^2} = x$
  
  Write $+x$ as the next term of the quotient.
- Multiply: Multiply $x$ by the divisor $x^2 + 4x - 1$ :
  
  $x(x^2 + 4x - 1) = x^3 + 4x^2 - x$
- Subtract: Subtract the result from the current polynomial:
  
  $\begin{aligned} &(x^3 + 4x^2 - 3x) \\ &\quad{}- (x^3 + 4x^2 - x) = -2x \end{aligned}$
- Bring Down: Bring down the next term ( $+1$ ) to get the temporary remainder:
  
  $-2x + 1$
Stop: The degree of the remainder ( $-2x+1$ , degree $1$ ) is less than the degree of the divisor ( $x^2+4x-1$ , degree $2$ ), so the division stops.
Result:
- The Quotient ( $H(x)$ ) is $4x^2 + x$ .
- The Remainder ( $S(x)$ ) is $-2x + 1$ .
Writing in Division Algorithm Form:

Based on the division algorithm, we can write the result as:
- Fraction Form:
  
  $\begin{aligned} &\frac{4x^4 + 17x^3 - 3x + 1}{x^2 + 4x - 1} \\ &= 4x^2 + x + \frac{-2x + 1}{x^2 + 4x - 1} \end{aligned}$
- Multiplication Form:
  
  $\begin{aligned} &4x^4 + 17x^3 - 3x + 1 \\ &= (x^2 + 4x - 1)(4x^2 + x) \\ &\quad{}+ (-2x + 1) \end{aligned}$

Long division records every quotient, multiplication, and subtraction step. That explicit layout works for divisors of any positive degree and makes the reconstruction $P(x)=Q(x)H(x)+S(x)$ easy to check.

Exercise

Find the quotient polynomial $H(x)$ and the remainder polynomial $S(x)$ after dividing $P(x) = x^3 - x + 9$ by $Q(x) = x^2 - 2x + 3$ . Confirm the result by multiplying the divisor with the quotient and adding the remainder.

State the result in the form $P(x) = Q(x) \cdot H(x) + S(x)$ .

Worked Solution

Complete $P(x)$ to become $x^3 + 0x^2 - x + 9$ .

\begin{array}{l} \qquad\qquad\qquad x + 2 \\ x^2-2x+3\overline{\big)x^3 + 0x^2 - x + 9} \\ \qquad\qquad\underline{-(x^3 - 2x^2 + 3x)} \\ \qquad\qquad\qquad\quad 2x^2 - 4x + 9 \\ \qquad\qquad\qquad\underline{-(2x^2 - 4x + 6)} \\ \qquad\qquad\qquad\qquad\qquad\qquad 3 \\ \end{array}

First divide the leading terms to obtain $x^3 \div x^2 = x$ . After multiplying and subtracting, the new leading term is $2x^2$ , so the next quotient term is $2x^2 \div x^2 = 2$ . The final remainder $3$ has lower degree than the divisor, so the division stops.

Quotient: $H(x) = x + 2$
Remainder: $S(x) = 3$
Division Algorithm Form:

$\begin{aligned} &x^3 - x + 9 \\ &= (x^2 - 2x + 3)(x + 2) + 3 \end{aligned}$

Preparing for Long Division

Before starting the division, there are a few things to prepare:

Order Terms:

Write the dividend ( $P(x)$ ) and the divisor ( $Q(x)$ ) in descending order of variable powers (from highest power to lowest).

Complete Terms:

If a power is missing, insert its zero coefficient as a placeholder. This keeps like powers in the same column during subtraction.

Example:

If $P(x) = 4x^4 + 17x^3 - 3x + 1$ , the $x^2$ term is missing.

So we write it as $P(x) = 4x^4 + 17x^3 + 0x^2 - 3x + 1$ .

Set Up Division:

Write the division in long division format, with $P(x)$ (the completed form) inside the division symbol and $Q(x)$ outside.

Steps for Long Division

The division process is performed step-by-step as follows:

Divide: Divide the first term of

P(x)

by the first term of

Q(x)

. Write the result as the first term of the quotient (

H(x)

) above the line.

Multiply: Multiply the quotient term just obtained by the entire divisor

Q(x)

Subtract: Write the result of the multiplication below

P(x)

, aligning like terms, then subtract it from

P(x)

to get a temporary remainder.

Bring Down: Bring down the next term from

P(x)

next to the temporary remainder to form a new polynomial.

Repeat: Repeat steps

1\text{-}4

with this new polynomial until the degree of the temporary remainder is less than the degree of the divisor

Q(x)

Divide

P(x) = 4x^4 + 17x^3 - 3x + 1

Q(x) = x^2 + 4x - 1

Preparation:

$P(x) = 4x^4 + 17x^3 + 0x^2 - 3x + 1$ (complete the $x^2$ term)
$Q(x) = x^2 + 4x - 1$

Division Process:

The quotient is $4x^2+x$ and the remainder is $-2x+1$ . The next steps show each multiplication and subtraction, including the zero coefficient of $x^2$ .

Step-by-Step Explanation:

Iteration $1$ :

Divide: Divide the first term $4x^4$ by the first term of the divisor $x^2$ :

$\frac{4x^4}{x^2} = 4x^2$

Write $4x^2$ as the first term of the quotient.
Multiply: Multiply $4x^2$ by the divisor $x^2 + 4x - 1$ :

$\begin{aligned} &4x^2(x^2 + 4x - 1) \\ &= 4x^4 + 16x^3 - 4x^2 \end{aligned}$
Subtract: Subtract the result from the initial polynomial:

$\begin{aligned} &(4x^4 + 17x^3 + 0x^2) \\ &\quad{}- (4x^4 + 16x^3 - 4x^2) \\ &= x^3 + 4x^2 \end{aligned}$
Bring Down: Bring down the next term ( $-3x$ ) to get the new polynomial:
$x^3 + 4x^2 - 3x$

Iteration $2$ :

Divide: Divide the first term of the new polynomial $x^3$ by the first term of the divisor $x^2$ :

$\frac{x^3}{x^2} = x$

Write $+x$ as the next term of the quotient.
Multiply: Multiply $x$ by the divisor $x^2 + 4x - 1$ :

$x(x^2 + 4x - 1) = x^3 + 4x^2 - x$
Subtract: Subtract the result from the current polynomial:

$\begin{aligned} &(x^3 + 4x^2 - 3x) \\ &\quad{}- (x^3 + 4x^2 - x) = -2x \end{aligned}$
Bring Down: Bring down the next term ( $+1$ ) to get the temporary remainder:

$-2x + 1$

Stop: The degree of the remainder ( $-2x+1$ , degree $1$ ) is less than the degree of the divisor ( $x^2+4x-1$ , degree $2$ ), so the division stops.

Result:

The Quotient ( $H(x)$ ) is $4x^2 + x$ .
The Remainder ( $S(x)$ ) is $-2x + 1$ .

Writing in Division Algorithm Form:

Based on the division algorithm, we can write the result as:

Fraction Form:

$\begin{aligned} &\frac{4x^4 + 17x^3 - 3x + 1}{x^2 + 4x - 1} \\ &= 4x^2 + x + \frac{-2x + 1}{x^2 + 4x - 1} \end{aligned}$
Multiplication Form:

$\begin{aligned} &4x^4 + 17x^3 - 3x + 1 \\ &= (x^2 + 4x - 1)(4x^2 + x) \\ &\quad{}+ (-2x + 1) \end{aligned}$

Long division records every quotient, multiplication, and subtraction step. That explicit layout works for divisors of any positive degree and makes the reconstruction

P(x)=Q(x)H(x)+S(x)

easy to check.

Exercise

Find the quotient polynomial

H(x)

and the remainder polynomial

S(x)

after dividing

P(x) = x^3 - x + 9

Q(x) = x^2 - 2x + 3

. Confirm the result by multiplying the divisor with the quotient and adding the remainder.

State the result in the form

P(x) = Q(x) \cdot H(x) + S(x)

Complete

P(x)

to become

x^3 + 0x^2 - x + 9

\begin{array}{l} \qquad\qquad\qquad x + 2 \\ x^2-2x+3\overline{\big)x^3 + 0x^2 - x + 9} \\ \qquad\qquad\underline{-(x^3 - 2x^2 + 3x)} \\ \qquad\qquad\qquad\quad 2x^2 - 4x + 9 \\ \qquad\qquad\qquad\underline{-(2x^2 - 4x + 6)} \\ \qquad\qquad\qquad\qquad\qquad\qquad 3 \\ \end{array}

First divide the leading terms to obtain

x^3 \div x^2 = x

. After multiplying and subtracting, the new leading term is

2x^2

, so the next quotient term is

2x^2 \div x^2 = 2

. The final remainder

3

has lower degree than the divisor, so the division stops.

Quotient: $H(x) = x + 2$

Remainder: $S(x) = 3$

Division Algorithm Form:

\begin{aligned} &x^3 - x + 9 \\ &= (x^2 - 2x + 3)(x + 2) + 3 \end{aligned}