What Is a Polynomial? Terms, Degree and the Factor Theorem

Q: Is a constant like 5 a polynomial?

Yes. It is a polynomial of degree 0, since 5 = 5x^0.

What counts as a polynomial and what does not, how to add, subtract and multiply them, what the degree tells you about the graph, and how the factor theorem finds factors from roots.

4 min read Supports 3 lessons

A polynomial is built from just three ingredients: numbers, a variable such as $x$ , and whole-number powers of it, combined with addition, subtraction and multiplication.

4x^3 - x + 7 \qquad x^2 - 5x + 6 \qquad 2x - 1 \qquad 9

All polynomials. A single number counts too.

\frac{1}{x} \qquad \sqrt{x} \qquad 2^x

Not polynomials: $\frac1x = x^{-1}$ and $\sqrt x = x^{1/2}$ have powers that are not whole numbers, and in $2^x$ the variable is in the exponent.

Degree and names

The degree is the highest power of $x$ . Polynomials are often named by degree: degree $1$ is linear, degree $2$ [[quadratic-formula|quadratic]], degree $3$ cubic. They are also named by how many terms they have: a monomial has one, a binomial two, a trinomial three.

Polynomials are usually written with the highest power first and the powers falling from left to right. The coefficient of that first, highest power is the leading coefficient.

A polynomial of degree $n$ has at most $n - 1$ turning points.

Adding and subtracting

Add polynomials by collecting like terms. To subtract, distribute the minus sign through the whole second bracket first.

$(3x^2 - 5x + 2) - (x^2 + 4x - 6)$
The minus sign applies to every term in the second bracket.

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Watch out

Forgetting to negate the last term of the second bracket is the classic error: $-(-6)$ is $+6$ .

Multiplying

Multiply every term of one polynomial by every term of the other, then collect like terms. A grid stops you missing a product. The degree of the result is the sum of the degrees.

13\times12 = 100 + 20 + 30 + 6 = 156

$(10 + 3)(10 + 2)$ works exactly like $(x + 3)(x + 2) = x^2 + 5x + 6$ .

Dividing, and the factor theorem

Polynomials divide like whole numbers, giving a quotient and a remainder. You rarely need long division, because two theorems do most of the work.

Remainder theorem

p(x) \div (x - a) \;\text{ leaves remainder }\; p(a)

Factor theorem

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

Worked example

Factorise $p(x) = x^3 - 2x^2 - 5x + 6$ completely.

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Try small values: $p(1) = 1 - 2 - 5 + 6 = 0$ , so $(x - 1)$ is a factor.

Tip

Whole-number roots must divide the constant term. For $x^3 - 2x^2 - 5x + 6$ , only $\pm1, \pm2, \pm3$ and $\pm6$ are worth trying.

Key idea

Each factor $(x - a)$ is a place where the graph crosses the $x$ -axis. Factorising a polynomial and finding where its graph meets the axis are the same job.

Common questions

Is a constant like 5 a polynomial?: Yes. It is a polynomial of degree $0$ , since $5 = 5x^0$ .
What does the degree of a polynomial tell you?: It is the highest power of the variable. It tells you the polynomial's basic shape and the most roots it can have: a degree $n$ polynomial has at most $n$ real roots.

Practise polynomials

The Adding & Subtracting Polynomials lesson has worked examples and unlimited practice that adapts to you.

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Common questions

Is a constant like 5 a polynomial?

Yes. It is a polynomial of degree $0$ , since $5 = 5x^0$ .

What does the degree of a polynomial tell you?

It is the highest power of the variable. It tells you the polynomial's basic shape and the most roots it can have: a degree $n$ polynomial has at most $n$ real roots.