Theorem

Factor and remainder theorem

Algebra • Polynomial • Factor and Remainder Theorem

Formula

(x - c) ∣ p (x) ⟺ p (c) = 0

For a polynomial

p (x)

and a constant

c

: the remainder on dividing

p (x)

(x - c)

p (c)

. In particular

(x - c)

is a factor of

p (x)

if and only if

p (c) = 0

How to use it

1To test $(x + 4)$ , set $x + 4 = 0$ and substitute $x = - 4$ ; the factor $(x + a)$ is tested at $x = - a$ .
2To find the remainder on division by $(x - c)$ , just evaluate $p (c)$ — no long division is needed.
3For a divisor $(a x - b)$ , substitute $x = \frac{b}{a}$ .

(x - 3)

a factor of

p (x) = x^{3} - 4 x^{2} + x + 6

Step-by-step

1
$p (3) = 27 - 36 + 3 + 6$
Why: The divisor is $(x - 3)$ , so evaluate at $x = 3$ .
2
$p (3) = 0$
Why: The remainder is zero, so the divisor goes in exactly.

Answer

Yes,

(x - 3)

is a factor

Watch out

It only tests LINEAR divisors. A zero remainder at one point says nothing about a quadratic factor, and

p (c) \neq = 0

rules out

(x - c)

but not other factors.

Also known as

factor theorem, remainder theorem, f(a)=0 root factor, polynomial division remainder, is (x-a) a factor