Procedure

Quotient on dividing by a linear factor

Algebra • Polynomial • Factor and Remainder Theorem

Statement

To divide

p (x) = a_{n} x^{n} + \dots + a_{0}

(x - c)

, write every coefficient down to the constant, including a

0

for each missing power. Bring the leading coefficient down unchanged, then repeatedly multiply the value just written by

c

and add it to the next coefficient. The first

n

values obtained are the coefficients of the quotient, in descending order, and the last is the remainder

p (c)

How to use it

1For a divisor $(x + k)$ the multiplier is $c = - k$ : dividing by $(x + 2)$ means working with $- 2$ .
2Write a $0$ for every absent power before starting — a skipped column shifts all later coefficients and is the usual way this goes wrong.
3The quotient's degree is one less than the dividend's, so count the values: a quartic divided by a linear factor leaves four quotient coefficients and one remainder.
4Equating coefficients in $p (x) = (x - c) \cdot q (x) + R$ is the same arithmetic written out longhand, and is a useful check.

Divide

x^{3} + 4 x - 5

(x - 1)

Step-by-step

1
$1, 0, 4, - 5$
Why: Coefficients of $p (x)$ with a $0$ inserted for the missing $x^{2}$ term.
2
$1; 0 + 1 (1) = 1; 4 + 1 (1) = 5; - 5 + 1 (5) = 0$
Why: Bring down the $1$ , then multiply by $c = 1$ and add, column by column.
3
$q (x) = x^{2} + x + 5, R = 0$
Why: The first three values are the quotient's coefficients; the last is the remainder.

Answer

x^{2} + x + 5

, with remainder

0

Watch out

It applies only to a MONIC linear divisor

(x - c)

. For

(a x - b)

divide by

(x - \frac{b}{a})

and then divide the whole quotient by

a

; the remainder is unaffected. A zero remainder confirms a factor, but a non-zero remainder does not invalidate the quotient.

Builds on

Also known as

synthetic division, polynomial long division, quotient on dividing a polynomial, divide polynomial by a linear factor, Horner's method