Polynomials, Division and the Factor Theorem

In short: Dividing a polynomial gives $P (x) = A (x) Q (x) + R (x)$ , where the remainder has a lower degree than the divisor. The remainder theorem says dividing by $(x - a)$ leaves $P (a)$ , and the factor theorem says $(x - a)$ is a factor exactly when $P (a) = 0$ .

Starting point: what you already know

A quadratic $a x^{2} + b x + c$ has a highest power of 2.
Expanding: $(x - 1) (x + 3) = x^{2} + 2 x - 3$ .
Factorising a quadratic and solving $a x^{2} + b x + c = 0$ .
Substitution: for $P (x) = x^{2} + 1$ , $P (2) = 5$ .
Long division of numbers: $47 \div 5 = 9$ remainder 2, so $47 = 5 \times 9 + 2$ .

The language of polynomials

A polynomial is a sum of terms with whole number powers of $x$ :

$P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$

Word	Meaning	In $P (x) = 2 x^{3} - 5 x + 7$
degree	highest power	3
leading term	term with the highest power	$2 x^{3}$
leading coefficient	number in front of it	2
constant term	term with no $x$	7
monic	leading coefficient is 1	not monic

A root or zero of $P$ is a number $α$ with $P (α) = 0$ . On the graph, it is where the curve meets the x-axis.

Expressions like $x$ or $\frac{1}{x}$ are not polynomials, because the powers are not whole numbers.

Dividing polynomials

Just like $47 = 5 \times 9 + 2$ , any polynomial can be divided by another:

$P (x) = A (x) \times Q (x) + R (x)$

where $Q (x)$ is the quotient and $R (x)$ is the remainder. The remainder has a lower degree than the divisor $A (x)$ .

The remainder theorem

Put $x = a$ into $P (x) = (x - a) Q (x) + R$ . The first part becomes zero, so:

Check the example: $P (1) = 1 - 2 + 4 - 5 = - 2$ . Same remainder, no division needed.

The factor theorem

If the remainder is zero, $(x - a)$ divides exactly.

Polynomials

Language

Degree and leading term
Monic, constant term
Roots are zeros

Division

P = A × Q + R
Remainder has lower degree

Remainder theorem

Remainder is P(a)

Factor theorem

P(a) = 0 means x - a is a factor
Test factors of the constant

Year 11Dividing by a quadratic

When dividing by a quadratic, the remainder can be linear, $R (x) = p x + q$ .

Example. Find the remainder when $P (x)$ is divided by $(x - 1) (x - 2)$ , given $P (1) = 3$ and $P (2) = 5$ . Write $P (x) = (x - 1) (x - 2) Q (x) + p x + q$ . Then $p + q = 3$ and $2 p + q = 5$ , so $p = 2$ , $q = 1$ . The remainder is $2 x + 1$ .

Long division by a quadratic works the same way, one leading term at a time. $x^{3} + 3 x^{2} + 5 \div (x^{2} + 2)$ : $x^{3} \div x^{2} = x$ , subtract $x^{3} + 2 x$ to leave $3 x^{2} - 2 x + 5$ ; then $3 x^{2} \div x^{2} = 3$ , subtract $3 x^{2} + 6$ to leave $- 2 x - 1$ . So the quotient is $x + 3$ and the remainder $- 2 x - 1$ .

Using values you already know. The remainder on dividing $x^{8}$ by $x^{2} - 1$ is $p x + q$ with $1 = p + q$ (at $x = 1$ ) and $1 = - p + q$ (at $x = - 1$ ), so the remainder is 1. If dividing by $x^{2} - 9$ leaves $2 x + 5$ , then dividing by $x - 3$ leaves $P (3) = 11$ , because the quadratic part is zero at $x = 3$ .

Odd and even polynomials. An odd polynomial has only odd powers ( $P (x) = a x^{3} + c x$ for a cubic); an even one has only even powers. Use this to cut down the unknowns before using the factor theorem.

Unknown inside the factor. If $x + k$ is a factor, then $P (- k) = 0$ . Factorise the resulting equation in $k$ and discard any value the question rules out.

Quick summary

A polynomial has whole number powers. The degree is the highest power.
$P (x) = A (x) Q (x) + R (x)$ , with the remainder of lower degree than $A (x)$ .
Remainder theorem: dividing by $(x - a)$ leaves remainder $P (a)$ .
Factor theorem: $(x - a)$ is a factor exactly when $P (a) = 0$ .
To factorise a cubic, test factors of the constant term, then divide.

Quiz

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What is the degree of $P (x) = 4 x^{5} - x^{2} + 9$ ?
1. A 4
2. B 5
3. C 9
4. D 2
Show answer
B: 5
The highest power of $x$ is 5.
What is the leading coefficient of $3 - 2 x + 7 x^{3}$ ?
1. A 3
2. B $- 2$
3. C 7
4. D 1
Show answer
C: 7
The highest power term is $7 x^{3}$ .
Which is a monic polynomial?
1. A $2 x^{2} + 1$
2. B $x^{3} - 4 x + 2$
3. C $5 x^{4} + x$
4. D $- x^{2} + 3$
Show answer
B: $x^{3} - 4 x + 2$
Monic means the leading coefficient is 1.