Polynomials

Recognising a polynomial

Consider the expression below.

Its three terms are $3x^2$, $-2x$ and $5$. The sign belongs to the term that follows it. The notation $P(x)$ names the polynomial and shows its input variable; it does not mean $P$ multiplied by $x$.

Here we use real-number coefficients. Fractions, decimals and irrational numbers may be coefficients. For example, $\frac12x^2+\sqrt{2}x-3$ is a polynomial in $x$: the restrictions concern the powers of $x$, not whether the coefficients are whole numbers. This follows the standard term-based definition used by OpenStax.

By contrast, $x^{-1}+2$, $\sqrt{x}+1$ and $2^x$ are not polynomials in $x$. They contain, respectively, a negative variable exponent, a fractional variable exponent, and a variable in the exponent. A denominator such as $2$ is allowed; a denominator containing $x$ introduces a different type of expression.

Terms, degree and standard form

Naming by the number of terms

After combining like terms, a polynomial with one non-zero term is a monomial, one with two is a binomial, and one with three is a trinomial. For example, $6x^2$ is a monomial, $x-4$ is a binomial, and $3x^2-2x+5$ is a trinomial.

Count after simplifying. The expression $x+2x$ is written as a sum, but it simplifies to the monomial $3x$. The number of terms and the degree describe different features.

Finding the degree

For a non-zero polynomial in one variable, the degree is the highest exponent whose coefficient is non-zero. The degree of $3x^2-2x+5$ is $2$, so it is quadratic. Degrees $1$ and $3$ are called linear and cubic.

In standard form, terms are arranged from the highest power to the lowest. The coefficient of the highest-degree term is the leading coefficient. In $5-2x+3x^2$, rearranging gives $3x^2-2x+5$, so the leading coefficient is $3$, not $5$.

A non-zero constant polynomial, such as $7$, has degree $0$. The zero polynomial is a special case: it has no highest non-zero term, and its degree is left undefined in this article. Do not label it degree zero simply because its value is zero.

Combining and subtracting polynomials

Like terms have the same variables raised to the same powers. Combine their coefficients while keeping the variable part unchanged.

The quadratic terms combine with quadratic terms, the linear terms with linear terms, and the constants with constants. You cannot combine $x^2$ and $x$ into a single like term.

When subtracting an entire polynomial, distribute the negative sign to every term inside the second set of parentheses.

This is the distributive property applied to a factor of $-1$. It is not a separate rule about changing signs at random.

Multiplication and area

Multiplying polynomials means multiplying every term in one factor by every term in the other, then combining like terms.

For positive $x$, a rectangle with side lengths $x+2$ and $x+3$ gives an area model. Splitting its sides creates four regions with areas $x^2$, $3x$, $2x$ and $6$. The two rectangular regions combine to $5x$ because their variable parts match.

The algebraic identity holds for every real $x$, even though physical lengths in this drawing must be positive. Reading the equation backwards gives a factorisation of $x^2+5x+6$. The Algebraic Identities page develops these relationships further.

Evaluating is not solving

To evaluate $P(x)=3x^2-2x+5$ at $x=2$, substitute the value:

That calculation finds an output. Solving $P(x)=0$ asks a different question: which inputs make the output zero? An expression does not become an equation until a condition is stated.

Polynomials can also involve several variables. In $2x^2y+3y-1$, the first term has total degree $3$, because $2+1=3$. To classify a polynomial, identify terms, combine like terms, arrange the powers and only then name the polynomial or its degree.