How do you determine the degree of a polynomial?

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ishpiro's profile pic

ishpiro | College Teacher | (Level 1) Educator

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To determine the degree of a polynomial, look for the term that has the highest exponent. That exponent is the degree of the polynomial.

A polynomial, by definition, is an algebraic expression containing sum of terms with whole (positive integer) exponents. For example,

`x^3 + 2x^2 - 5x + 7` is a polynomial of third degree, or cubic polynomial, because the highest exponent is 3.

Similarly,

`2x - 7x^2 + x^5 - 1` would be the polynomial of fifth degree, because the highest exponent is 5. Notice that this polynomial is not written in standard form, where the exponents are arranged in descending order. It usually is helpful to rewrite the polynomials in standard form. In this case, it would be

`x^5 - 7x^2 + 2x - 1` .

When you cannot see any exponents in the polynomial, that means that it either of degree 1, as in

`2x - 7` (Here, in the term 2x, x has the exponent of 1, which is implied but not written.) This is a first degree, or linear polynomial.

Or, it could be 0, as in

`5` . There is no x present, which means that x has exponent 0. So, this is a polynomial (or, rather, a monomial) of  degree zero.

If a polynomial has more than one variable, which you will not see very often, you would have to find the term with the highest sum of exponents of both variables. For example,

`a^2b - ab + a^2` has degree three, because the highest sum of exponents is 3, in the term `a^2b` : 2 + 1 = 3.

Hope this helps.

mathace's profile pic

mathace | (Level 3) Assistant Educator

Posted on

You determine the degree of the polynomial by finding the highest exponent.

For example in the polynomial `x^2+3x-5`

the highest exponent or power is 2. Therefore 2 is the degree of the equation.

For example in the polynomial `2x^3-4x^2+5x-1`

the highest exponent or power is 3. Therefore 3 is the degree of the equation.

For example `x-5=x^1-5`

the highest exponent or power is 1. Therefore 1 is the degree of the equation.

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