The degree of a polynomial is the maximum of the degrees of all terms in the polynomial. For example, in 2x³ + 4x² + x + 7, the term of highest degree is 2x³; this term, and therefore the entire polynomial, are said to have degree 3.

Degree (or order) can also describe the power to which a single term in a polynomial (or any function) is raised. The number is the effective exponent, so it is the effect of a variable after the equation is simplified.

Examples

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• The equation $3\; -\; 5\; x\; +\; 2\; x^5\; -\; 7\; x^9$ has degree 9.
• The equation $(y\; -\; 3)(2y\; +\; 6)(-4y\; -\; 21)$ has degree 3.
• The equation $(3\; z^8\; +\; z^5\; -\; 4\; z^2\; +\; 6)\; +\; (-3\; z^8\; +\; 8\; z^4\; +\; 2\; z^3\; +\; 14\; z)$ has degree 5.
• In the equation $y\; =\; x\; =\; x^1$, the variable x is in the first order.
• In the equation $f(x)\; =\; x^3\; +\; x$, x is present in first and third order.
• In the equation $z\; =\; t^4\; +\; 3\; \backslash cdot\; t^2$, the 4th order coefficient of t is 1, and the second order coefficient is 3. The third and 1st order coefficients are said to be zero.
• In the equation (not technically a polynomial) $v\text{'}\; =\; \backslash frac\{1\}\{x^3\}$, x is in the negative third order.
• In the equation (again not stricly a polynomial) $y\; =\; \backslash sqrt*\{x^4\; +\; x\}$, x is present in the first order and negative second order.
• When we define a number $C\; \backslash approx\; 3\; \backslash cdot\; 10^8\; \backslash frac\{\backslash mathrm\{meters\}\}\{\backslash mathrm\{second\}\}$, 10 is 8th order and the order of magnitude of C is 8.

Behaviour under addition, subtraction and multiplication

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The degree of the sum (or difference) of two polynomials is equal to or less than the greater of their degrees i.e.

$deg(P\; \backslash pm\; Q)\; \backslash leq\; max(deg(P),deg(Q))$.

For example:

• The degree of $(x^3+x)+(x^2+1)$ is 3.
• The degree of $(x^3+x)-(x^3+x^2)$ is 2, because the terms in x³ cancel out.

The degree of the product of two polynomials is the sum of their degrees (provided the polynomials' coefficients belong to an integral domain) i.e.

$deg(PQ)\; =\; deg(P)\; +\; deg(Q)$.

For example:

• The degree of $(x^3+x)(x^2+1)$ is 5.
• The degree of $(x^3+x)(x^3+x^2)$ is 6.

This property can be used to show that an extension of the Euclidean algorithm applies to polynomials, and so (with appropriate assumptions about the ring to which the polynomials' coefficients belong) polynomials form an Euclidean domain - see Polynomial ring for more details.

Extension to polynomials with two or more variables

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For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x²y² + 3x³ + 4y has degree 4, the same degree as the term x²y².

See also

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• Degree (mathematics) — other meanings of degree in mathematics

Grado (polinomio) | graad (polynoom) Stopień wielomianu

Polynomials