medium difficulty8 min

Polynomial — Definition and Degree

A polynomial is an algebraic expression made using variables, constants, coefficients, and non-negative whole number powers of variables. The degree is the highest power of the variable present in the polynomial.

In one line: Degree means highest power, not biggest number.

Practice This Concept

Understand it

In one variable, expressions like 3x + 2, x² - 5x + 6, and 7 are polynomials because the powers of x are whole numbers. Expressions with variables in the denominator or square root of the variable are not polynomials at this level. The degree tells the type of polynomial, such as linear for degree 1 and quadratic for degree 2.

Example

In 5x³ - 2x + 8, the highest power of x is 3, so the degree is 3. In 4x + 9, the degree is 1, so it is linear.

Formula and rule

Polynomial rule: powers of variables must be non-negative whole numbers.

Degree in one variable is the highest exponent of the variable.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the degree of 9 - 3x + 2x⁴.

1The powers of x are 0 for 9, 1 for -3x, and 4 for 2x⁴.
2The highest power is 4, so the degree is 4.

Method you can reuse

Check each term, note the power of the variable, ensure the powers are non-negative whole numbers, and then choose the highest power as the degree.

Visual tip

A term table can be used with columns for term, coefficient, power of variable, and role in expression.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the degree of the polynomial 6x² - 4x + 1?

Think first, then show answer

The degree is 2 because the highest power of x in 6x² - 4x + 1 is 2.

Common wrong answer

For 7x³ + 2x² + 5, a common wrong answer is degree 7 because students look at the largest coefficient instead of the largest exponent.

Don't mix this up

Students sometimes count the number of terms as the degree. For example, 2x + 7 has two terms, but its degree is 1, not 2.

Exam tip

First arrange mentally by powers of the variable, then identify the highest exponent. Do not use the coefficient to decide the degree.