Polynomials

Reviewed by the Helping with Math Team · Updated 14 October 2026

What will I learn from this article?

After reading this article, you will be able to:

• define a polynomial;
• name the parts of a polynomial;
• know the standard form of polynomials;
• classify a polynomial by its degree;
• name a polynomial based on the number of terms;
• add, subtract, multiply and divide polynomials;
• use the main properties of polynomials; and
• see where polynomials are used

What is a polynomial?

A polynomial is an algebraic expression made of one or more terms joined by + or −. Each term is a number (the coefficient) times one or more variables raised to whole-number powers: 0, 1, 2, 3 and so on.

For example, 2x + 3 is a polynomial, and so is 3x² + 8x + 1. Even a single term such as 2x, or a single number such as 7, counts as a polynomial.

In one variable x, every polynomial can be written in this general form:

$a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$

Here a_n, a_n−1, …, a₀ are numbers called coefficients, and n is a whole number. The letter x is the variable (sometimes called the indeterminate): it does not stand for one fixed value, and any number can be substituted for it.

Not polynomials: 1/x (a variable in the denominator), √x (a variable under a root), x⁻² (a negative exponent) and x^1/2 (a fraction exponent).

x³

0

x²

1

x

-2

number

-3

x² − 2x − 3 has degree 2 (quadratic) and 3 terms (a trinomial).

What are the parts of a polynomial?

The diagram below labels the parts of the polynomial 4x² + 3x − 7.

• A coefficient refers to a number that is being multiplied by the variable.
• A variable is a symbol or letter that works as a placeholder for a quantity that may assume any given value.
• An exponent (power) tells how many times the variable is multiplied by itself: x² means x × x.
• An operator is a symbol that indicates an operation to be performed, such as addition, subtraction, multiplication, etc.
• A constant term is a value or quantity that does not change. It is the fixed value in a mathematical statement.
• A term is one part of the polynomial between the + and − signs. It can be a number, a variable, or a number times variables.

What is the standard form of a polynomial?

The standard form of a polynomial refers to the writing of a polynomial in which the exponent of variables is arranged in descending order. The table below shows polynomials in non-standard and standard form.

What is the degree of a polynomial?

The degree of a polynomial in one variable is the highest exponent of the variable in any term with a non-zero coefficient. For example, 4x³ + 2x − 9 has degree 3. (With more than one variable, add the exponents in each term and take the largest: x²y³ + x has degree 5.)

Polynomials are often named by their degree:

• Constant polynomial (degree 0): just a number, such as 8.
• Linear polynomial (degree 1): such as 2x or x + 3.
• Quadratic polynomial (degree 2): such as 6x² + x + 9.
• Cubic polynomial (degree 3): such as 10x³ + 2x² + 5x + 1.
• Quartic polynomial (degree 4): such as 7x⁴ + 3x³ + 9x² + 10x + 1.

The number 0 on its own is the zero polynomial. It is a special case: most books say its degree is undefined, so it is not the same as a constant polynomial like 8.

To summarize the different degrees of polynomial, look at the table below.

Monomials, binomials and trinomials

The terms of a polynomial are the parts separated by + and − signs.

Polynomials with one, two or three terms have special names: monomial, binomial and trinomial. A polynomial with more terms is just called a polynomial.

• Monomial1 term
• Binomial2 terms
• Trinomial3 terms

Monomial

A monomial is a polynomial or an algebraic expression that contains exactly one term. For a polynomial to be referred to as a monomial, the single term should be a non-zero term. Some examples of monomials are:

• x
• 8x
• 3
• 6x³
• -10x

Binomial

A binomial is a polynomial with exactly two terms. It is the sum or difference of two monomials (that are not like terms). A few examples of binomial are:

• 2x + 3
• 4x² + 4
• 7x⁴ + 3y²
• -x⁷ + 9xy³

Trinomial

A trinomial is an expression which is composed of exactly three terms. Some examples of trinomial are:

• x³ + 2x + 1
• 8x⁶ + 3x² + 2
• -4x⁵ + x³ + 9

To summarize the different classifications of polynomials, observe the table below.

What are the properties of polynomials?

These are the properties of polynomials you will use most. The first four are the same rules you know for numbers.

1. Closure: adding, subtracting or multiplying two polynomials always gives another polynomial. (Dividing does not always: (x + 1) ÷ x is not a polynomial.)
2. Commutative property: the order does not change a sum or product. P + Q = Q + P and P × Q = Q × P.
3. Associative property: the grouping does not change a sum or product. (P + Q) + R = P + (Q + R).
4. Distributive property: P × (Q + R) = P × Q + P × R. This is how you multiply polynomials.
5. Degree rules: the degree of P + Q or P − Q is at most the larger of the two degrees (it can be smaller if the top terms cancel). The degree of P × Q is the degree of P plus the degree of Q.
6. Division algorithm: when P(x) is divided by D(x), you get a quotient Q(x) and a remainder R(x) with P(x) = D(x) × Q(x) + R(x), where R(x) has a lower degree than D(x).
7. Remainder theorem: when P(x) is divided by (x − a), the remainder is P(a).
8. Factor theorem: (x − a) is a factor of P(x) exactly when P(a) = 0. (This is also called Bézout’s theorem for polynomials.)
9. Number of roots: a polynomial of degree n ≥ 1 has at most n different roots (values of x that make it zero).
10. Fundamental theorem of algebra: every polynomial of degree 1 or more has at least one complex root. Counted with repeats, a degree n polynomial has exactly n complex roots.
11. Complex roots come in pairs: if a polynomial has real coefficients and a + bi is a root, then a − bi is also a root, and x² − 2ax + a² + b² is a factor.
12. Descartes’ rule of signs: the number of positive real roots equals the number of sign changes in the coefficients, or is less than it by an even number.
13. Intermediate value theorem: if P(a) and P(b) have opposite signs, P(x) has at least one root between a and b, because a polynomial graph has no gaps or jumps.

Adding, subtracting, multiplying and dividing polynomials

There are four primary polynomial operations namely, addition of polynomials, subtraction of polynomials, multiplication of polynomials, and division of polynomials.

Addition of Polynomials

When adding polynomials, add like terms: terms with the same variable raised to the same power. 3x² and 5x² are like terms; 3x² and 3x are not.

To add polynomials:

1. Write each polynomial in standard form (highest power first).
2. Line up like terms in columns (or group them in a row).
3. Add the coefficients in each column. The variable and exponent stay the same.

Subtraction of Polynomials

To subtract a polynomial, add its opposite: change the sign of every term in the polynomial being taken away (the subtrahend), then add like terms as usual.

To subtract polynomials:

1. Write both in standard form.
2. Change the sign of every term in the polynomial you are subtracting.
3. Combine like terms.

Multiplication of Polynomials

To multiply polynomials:

1. Multiply every term of the first polynomial by every term of the second (the distributive property).
2. Use the exponent rule x^a × x^b = x^a+b: when you multiply powers of the same variable, add the exponents.
3. Combine like terms.

Division of Polynomials

Dividing one polynomial by another gives a quotient and sometimes a remainder. The method is the same as long division with numbers.

1. Write both polynomials in standard form.
2. Divide the first term of the dividend by the first term of the divisor. Write the answer on top.
3. Multiply the whole divisor by that answer, write it underneath and subtract.
4. Bring down the next term and repeat until the remainder has a lower degree than the divisor.

When you divide powers of the same variable, subtract the exponents: x^a ÷ x^b = x^a−b.

Example

What is the quotient when x² + 5x + 6 is divided by x + 3?

Solution

Step 1: x² ÷ x = x. Write x on top. Multiply: x(x + 3) = x² + 3x. Subtract: (x² + 5x) − (x² + 3x) = 2x. Bring down the 6 to get 2x + 6.

Step 2: 2x ÷ x = 2. Write + 2 on top. Multiply: 2(x + 3) = 2x + 6. Subtract: (2x + 6) − (2x + 6) = 0.

Therefore, the quotient is x + 2 with remainder 0. Check: (x + 3)(x + 2) = x² + 5x + 6.

Where are polynomials used?

Polynomials appear across math, science and everyday life. For instance, they are used to:

• construct polynomial equations, which encode a wide variety of problems ranging from elementary word problems to complex scientific problems;
• define polynomial functions, which appear in a variety of contexts ranging from elementary chemistry and physics to economics and social science; and
• approximate other functions in calculus and numerical analysis.

Polynomials are used in advanced mathematics to construct polynomial rings and algebraic varieties, both of which are fundamental concepts in algebra and algebraic geometry.

Related lessons: classification, coefficient, division of polynomials, factor theorem, long division, word problems, multiplication, standard form and subtraction. More practice: division of polynomials worksheets and multiplication of polynomials worksheets.

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