SPI Mathematics · Polynomials

Division Algorithm, Remainder and Factor Theorems

Dividing p(x) by g(x) gives a quotient and a remainder of lower degree than g(x); the remainder on dividing by x − a is p(a), and x − a is a factor exactly when p(a) = 0.

All Polynomials notes

Why this matters

Ten questions in the bank: two EASY, six MODERATE and two HARD, the hardest block in the chapter. Long division gives the quotient, but most of these are faster by the remainder theorem: put one number into the polynomial instead of dividing.

Concept 1 of 3: The division algorithm for polynomials

Polynomial division works like long division of numbers. At each step divide the leading term by the divisor's leading term, multiply back, subtract, and carry on until what is left has a lower degree than the divisor. That leftover is the remainder.

Definition

$p(x) = g(x) \times q(x) + r(x)$ : dividend $=$ divisor $\times$ quotient $+$ remainder.
The remainder has a lower degree than the divisor (or is 0).
Degree of the quotient $=$ degree of the dividend $-$ degree of the divisor.
Write a missing power with a 0 coefficient before dividing, so the columns line up.
Divisible means the remainder is 0. If $g(x)$ divides $p(x)$ , you can write $p(x) = g(x) \times (\text{unknown quotient})$ and compare coefficients.

Division algorithm

p(x) = g(x)\,q(x) + r(x), \quad \deg r < \deg g \qquad \deg q = \deg p - \deg g

Worked example

Divide

2x^3 + 3x^2 - x + 5

x + 2

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Polynomials · Division Algorithm, Remainder and Factor Theorems

On dividing the polynomial

6x^3 - 11x^2 + 7x + 5

2x - 3

, the quotient is

Skipping a missing power

x^3 - 7x + 6

has no

x^2

term. Write it as

x^3 + 0x^2 - 7x + 6

before dividing, or the subtraction lines up the wrong columns.

Expecting a number as every remainder

The remainder only has to be of lower degree than the divisor. Divided by a quadratic, the remainder can be linear, such as

3x + 1

Drill 5 more on the division algorithm for polynomials

Concept 2 of 3: The remainder theorem

Put

x = a

into

p(x) = (x - a)q(x) + r

. The first part becomes 0, so

p(a) = r

. The remainder on dividing by

x - a

is just the value of the polynomial at a, with no division needed.

Definition

The remainder when $p(x)$ is divided by $x - a$ is $p(a)$ .
For a divisor $x + a$ , put $x = -a$ . For $ax - b$ , put $x = \frac{b}{a}$ .
The number to subtract to make $p(x)$ exactly divisible by $x - a$ is the remainder $p(a)$ .

Remainder theorem

\text{remainder of } p(x) \div (x - a) = p(a)

Worked example

Find the remainder when

x^3 - 3x^2 + 4x - 5

is divided by

x - 2

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Polynomials · Division Algorithm, Remainder and Factor Theorems

The real number which should be subtracted from the polynomial

2x^3 - x^2 - 13x - 6

so that the polynomial

x - 3

divides it exactly is

Putting in the wrong sign

For the divisor

x + 3

, put

x = -3

, not

x = 3

. The number you put in is the one that makes the divisor 0.

Concept 3 of 3: The factor theorem

A factor divides with remainder 0. By the remainder theorem that remainder is

p(a)

, so

x - a

is a factor exactly when

p(a) = 0

, that is, when a is a zero.

Definition

$x - a$ is a factor of $p(x)$ when $p(a) = 0$ , and only then.
$x + a$ is a factor when $p(-a) = 0$ .
An unknown coefficient: put the zero in, set the value to 0 and solve.
Two zeroes known, say $\sqrt{3}$ and $-\sqrt{3}$ : then $(x - \sqrt{3})(x + \sqrt{3}) = x^2 - 3$ is a factor. Divide by it to find the rest.

Factor theorem

(x - a) \text{ is a factor of } p(x) \text{ when } p(a) = 0

Worked example

Find k if

x - 1

is a factor of

x^3 + kx^2 - 3x + 4

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Polynomials · Division Algorithm, Remainder and Factor Theorems

(x + a)

is a factor of

2x^2 + 2ax + 5x + 10

then a =

Mixing up the factor and the zero

The factor

x + 5

goes with the zero

-5

, and the factor

x - 5

with the zero 5. Test a factor by putting in the number that makes it 0.

Drill 2 more on the factor theorem

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

The division algorithm for polynomials
Division algorithm
$p(x) = g(x)\,q(x) + r(x), \quad \deg r < \deg g \qquad \deg q = \deg p - \deg g$
The remainder theorem
Remainder theorem
$\text{remainder of } p(x) \div (x - a) = p(a)$
The factor theorem
Factor theorem
$(x - a) \text{ is a factor of } p(x) \text{ when } p(a) = 0$

Watch out for (4)

Skipping a missing power
→ The division algorithm for polynomials
Expecting a number as every remainder
→ The division algorithm for polynomials
Putting in the wrong sign
→ The remainder theorem
Mixing up the factor and the zero
→ The factor theorem

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Sit an SPI past paper Drill all 10 questions