The remainder theorem and the factor theorem

Warm-up

Three quick questions from earlier lessons.

Discover

$p(x) = x^2 - 5x + 6$ . Work out $p(2)$ . What does the answer tell you?

$p(2) = 0$ , so $x - 2$ is a factor
$p(2) = 0$ , so $x + 2$ is a factor

Key idea

Dividing by $x - r$ and evaluating at $r$ are linked.

Remainder theorem: the remainder of

p(x) \div (x - r)

is

p(r)

. Factor theorem:

x - r

is a factor of

p(x)

exactly when

p(r) = 0

.

Worked example

Find the remainder of $p(x) = x^3 - 4x + 1$ divided by $x - 2$ .

Remainder

= p(2)

Remainder theorem: evaluate p at 2, not at −2.

p(2) = 1

8 − 8 + 1.

So

x - 2

is not a factor

The remainder is not 0.

Your turn, guided

Your turn: is $x - 3$ a factor of $p(x) = x^3 - 27$ ?

p(3) = 27 - 27

Factor theorem: compute p(3).

p(3) = 0

, so yes

Type the value of p(3).

Practice ★

What is the remainder when $p(x) = x^3 - 5x + 7$ is divided by $x - 3$ ?
What is the remainder when $p(x) = x^3 - 2x + 6$ is divided by $x - 3$ ?
What is the remainder when $p(x) = x^3 - 7x + 3$ is divided by $x - 2$ ?
What is the remainder when $p(x) = x^3 - 1x + 6$ is divided by $x - 4$ ?

Watch out!

Practice ★★

Is $x - 3$ a factor of $p(x) = x^2 - 7x + 12$ ?
1. Yes, because $p(3) = 0$
2. No, because $p(-3) \ne 0$
3. Only if dividing $p(x)$ by $x - 3$ leaves the remainder 3
Is $x - 5$ a factor of $p(x) = x^2 - 8x + 15$ ?
1. Yes, because $p(5) = 0$
2. No, because $p(-5) \ne 0$
3. Only if dividing $p(x)$ by $x - 5$ leaves the remainder 5
Is $x - 1$ a factor of $p(x) = x^2 - 6x + 5$ ?
1. Yes, because $p(1) = 0$
2. No, because $p(-1) \ne 0$
3. Only if dividing $p(x)$ by $x - 1$ leaves the remainder 1

For which value of $k$ is $x - 2$ a factor of $x^2 + kx + 4$ ?
For which value of $k$ is $x - 1$ a factor of $x^2 + kx + 2$ ?

Exit ticket

Three questions to check you have got it.

What is the remainder when $p(x) = x^3 - 5x + 7$ is divided by $x - 3$ ?
What is the remainder when $p(x) = x^3 - 2x + 6$ is divided by $x - 3$ ?
What is the remainder when $p(x) = x^3 - 7x + 3$ is divided by $x - 2$ ?

Summary card

Remainder and factor theorems

Remainder of $p(x) \div (x - r)$ : $p(r)$ .
$x - r$ is a factor $\iff p(r) = 0$ .
For $x + r$ , evaluate at $-r$ .

Watch out: a factor needs a remainder of $0$ .

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