Polynomials and the Factor Theorem

In short: A polynomial is a sum of terms like $a x^{n}$ with whole number powers, and its degree is the highest power. By the factor theorem, if $P (a) = 0$ then $(x - a)$ is a factor, and by the remainder theorem dividing $P (x)$ by $(x - a)$ leaves $P (a)$ .

Starting point: Years 9 and 10

Polynomial words

A polynomial is a sum of terms $a x^{n}$ where each power $n$ is a whole number $0, 1, 2, \dots$

For $P (x) = 2 x^{3} - 5 x^{2} + x - 7$ :

The degree is the highest power: 3. So it is a cubic.
The leading term is $2 x^{3}$ and the leading coefficient is 2.
The constant term is $- 7$ .

Degree 1 is linear, 2 is quadratic, 3 is cubic, 4 is quartic. $\frac{1}{x}$ and $x$ are not polynomials, because their powers are not whole numbers.

Evaluating and expanding

$P (2)$ means substitute $x = 2$ : $P (2) = 16 - 20 + 2 - 7 = - 9$ .

To expand, multiply every term by every term and collect like terms:

$(x + 2) (x^{2} - 3 x + 1) = x^{3} - 3 x^{2} + x + 2 x^{2} - 6 x + 2 = x^{3} - x^{2} - 5 x + 2$

Dividing polynomials

Long division works like number long division: divide the leading terms, multiply back, subtract, bring down, repeat.

The remainder theorem

When $P (x)$ is divided by $x - a$ , the remainder is $P (a)$ .

Check: $P (2) = 8 + 8 - 10 - 6 = 0$ , matching the remainder above.

The factor theorem

Finding all the zeros

Try small factors of the constant term: $\pm 1, \pm 2, \pm 3, \dots$ until $P (a) = 0$ .
Divide by $x - a$ to get a quadratic.
Factorise the quadratic.

From the example, $x^{2} + 4 x + 3 = (x + 1) (x + 3)$ , so

$x^{3} + 2 x^{2} - 5 x - 6 = (x - 2) (x + 1) (x + 3)$

The zeros are $x = 2$ , $- 1$ and $- 3$ .

Cubic graphs

$y = a (x - h)^{3} + k$ is $y = x^{3}$ moved right $h$ and up $k$ . Its flat point of inflection is at $(h, k)$ . A negative $a$ flips it.

A factorised cubic like $y = (x - 2) (x + 1) (x + 3)$ crosses the x-axis at each zero. With a positive leading coefficient it starts low on the left and ends high on the right.

Polynomials

Language

Degree, leading term
Constant term

Operations

Expand
Long division

Theorems

Remainder: P(a)
Factor: P(a) = 0

Zeros

Try factors of constant
Divide, then factorise

Graphs

y = a(x - h)³ + k
Cross at single zeros
Touch at double zeros

Year 11Sketching polynomials

For $y = P (x)$ in factorised form:

A single zero: the graph cuts straight through.
A double zero, from $(x - a)^{2}$ : the graph touches the axis and turns back.
A triple zero, from $(x - a)^{3}$ : it flattens and passes through.
The y-intercept is $P (0)$ .
The ends follow the leading term. For an even degree with $a > 0$ , both ends go up. For an odd degree with $a > 0$ , it goes from bottom left to top right.

Example. $y = - x (x - 3)^{2}$ : zeros at 0 (cut) and 3 (touch), leading term $- x^{3}$ , so it comes down from top left and finishes bottom right.

Equations. A polynomial equation of degree $n$ has at most $n$ real solutions.

Degree of a product. Multiplying polynomials adds their degrees: a cubic times a quadratic has degree 5.

Equating coefficients. If two polynomials are equal for all $x$ , the coefficients of each power match. For $3 x^{2} + 5 x + 1 = a (x + 1)^{2} + b (x + 1) + c$ : expand the right side to $a x^{2} + (2 a + b) x + (a + b + c)$ , so $a = 3$ , $2 a + b = 5$ gives $b = - 1$ , and $a + b + c = 1$ gives $c = - 1$ . (Substituting handy values of $x$ , like $x = - 1$ , also works.)

Building from the roots. A monic cubic with roots $- 2$ , $0$ and $2$ is $y = x (x + 2) (x - 2) = x^{3} - 4 x$ .

Two unknowns, two factors. If $x - 1$ and $x + 3$ are both factors of $P (x) = x^{3} + a x^{2} + b x - 3$ , then $P (1) = 0$ and $P (- 3) = 0$ give two equations: $a + b = 2$ and $9 a - 3 b = 30$ . Solving gives $a = 3$ , $b = - 1$ .

Odd and even polynomials. A polynomial with only odd powers is odd, and with only even powers (and a constant) is even. If an odd polynomial has a root at $x = 2$ , it also has one at $x = - 2$ .

Watch the leading term. In $y = x^{2} (x - 2) (1 - x)$ , the factor $(1 - x)$ contributes $- x$ , so the leading term is $- x^{4}$ and both ends go down.

Quick summary

A polynomial is a sum of terms $a x^{n}$ with whole number powers. The degree is the highest power.
Long division: divide leading terms, multiply, subtract, repeat.
Remainder theorem: dividing by $x - a$ leaves $P (a)$ .
Factor theorem: $P (a) = 0$ means $x - a$ is a factor.
Find zeros by testing factors of the constant term, dividing, then factorising.

Quiz

13 questions with instant marking. It is free, and your scores are saved to your account so you can carry on from any device.

Turn on JavaScript to sign in and take the quiz.

Try 3 questions now

What is the degree of $5 x^{4} - 2 x^{2} + 7$ ?
1. A 2
2. B 4
3. C 5
4. D 7
Show answer
B: 4
The highest power is 4.
Which is NOT a polynomial?
1. A $x^{3} - 1$
2. B $7$
3. C $x^{2} + \frac{1}{x}$
4. D $3 x - 2$
Show answer
C: $x^{2} + \frac{1}{x}$
$\frac{1}{x} = x^{- 1}$ has a negative power.
What is the leading coefficient of $- 3 x^{3} + x - 4$ ?
1. A $- 3$
2. B 3
3. C $- 4$
4. D 1
Show answer
A: $- 3$
The number in front of the highest power.