Degree of a polynomial

In mathematics, the degree of a polynomial shows us how complex the polynomial is. It is the largest power or exponent in any term of the polynomial. Each part of a polynomial, called a monomial, has its own degree. This degree is the total of the exponents of all the variables in that part.

For example, in the term ( x^2y^3 ), the degree is 5 because we add the exponents 2 and 3 together.

The degree of a polynomial helps us understand how it behaves, especially when the variable gets very large. For a polynomial with one variable, like ( x^4 + 3x^2 + 1 ), the degree is the largest exponent, which is 4 here.

Sometimes, polynomials look complicated but can be simplified. For example, the expression ( (x + 1)^2 - (x - 1)^2 ) looks tricky, but when we expand it, we find it simplifies to ( 4x ), which has a degree of 1. This shows that even when polynomials seem complex, we can still find their degree by simplifying them. Knowing the degree of a polynomial is important in many areas of math, from algebra to calculus.

Names of polynomials by degree

Polynomials are named based on their degree. The degree is the largest exponent in the polynomial. For example, a polynomial with degree 2 is called quadratic. One with degree 3 is called cubic. Here are some common names:

• Degree 0 – non-zero constant
• Degree 1 – linear
• Degree 2 – quadratic
• Degree 3 – cubic
• Degree 4 – quartic (or biquadratic if all terms have even degree)
• Degree 5 – quintic

These names help us describe how complex a polynomial is. For higher degrees, the names follow Latin numbers and end in -ic.

Examples

Polynomials are expressions made from variables and numbers using addition, subtraction, and multiplication. To find the degree of a polynomial, we look for the term with the highest exponent.

For example, the polynomial ((y - 3)(2y + 6)(-4y - 21)) simplifies to (-8y^{3} - 42y^{2} + 72y + 378). The highest exponent here is 3, so this is a cubic polynomial.

Another example is ((3z^{8} + z^{5} - 4z^{2} + 6) + (-3z^{8} + 8z^{4} + 2z^{3} + 14z)). When we combine like terms, the (z^{8}) terms cancel out, leaving (z^{5} + 8z^{4} + 2z^{3} - 4z^{2} + 14z + 6). The highest exponent now is 5, making this a quintic polynomial.

Behavior under polynomial operations

The degree of a polynomial tells us the highest power of the variable in that polynomial. When we add, multiply, or compose polynomials, the degree of the result relates to the degrees of the original polynomials in specific ways.

When we add two polynomials, the degree of the sum is at most the larger of the two degrees. For example, adding (x^3 + x) and (x^2 + 1) gives (x^3 + x^2 + x + 1), which has degree 3, the larger of the two original degrees.

When we multiply two polynomials, the degree of the product is the sum of the degrees of the two polynomials. For instance, multiplying (x^3 + x) and (x^2 + 1) results in (x^5 + 2x^3 + x), which has degree 5, equal to (3 + 2).

Degree of the zero polynomial

The zero polynomial, which is just the number 0, is a special case in math. Because it has no terms with numbers other than zero, we usually say it has no degree.

To keep math rules simple, some people say its degree is negative infinity.

This helps keep things consistent. For example, adding the zero polynomial to another polynomial doesn’t change the degree of the other polynomial. And multiplying any polynomial by the zero polynomial always gives the zero polynomial, which fits with the idea that its degree is negative infinity.

Computed from the function values

We can find the degree of a polynomial by watching how the function changes when the input values become very large. One way uses a special kind of math called asymptotic analysis. This looks at how the function grows compared to a simple power of the input.

This method can also help us understand the "degree" of some functions that are not polynomials. For example, the multiplicative inverse (1/x) has a degree of -1. The square root function has a degree of 1/2. The logarithm function has a degree of 0. And the exponential function has a degree of infinity.

Extension to polynomials with two or more variables

When we have polynomials with two or more variables, we find the degree of each term by adding up the exponents of the variables in that term. The degree of the whole polynomial is the largest of these totals. For example, in the polynomial (x^{2}y^{2} + 3x^{3} + 4y), the term (x^{2}y^{2}) has a degree of 4 (because (2 + 2 = 4)), making the degree of the whole polynomial 4.

A polynomial with two variables can also be viewed in different ways: as a polynomial in one variable with coefficients that are polynomials in the other variable. For instance, the same polynomial (x^{2}y^{2} + 3x^{3} + 4y) can be seen as a polynomial in (x) with coefficients that involve (y), or as a polynomial in (y) with coefficients that involve (x). In this case, it has a degree of 3 in (x) and a degree of 2 in (y).

Degree function in abstract algebra

When we talk about polynomials in a special kind of math called a field, the degree of a polynomial works in a useful way. If you multiply two polynomials, the degree of the answer is just the sum of their degrees. This helps mathematicians study these polynomials more easily.

But this can change if we use other math systems that are not fields. For example, in a system where numbers are only looked at modulo 4, multiplying two degree-1 polynomials can sometimes give a result of degree 0. This shows why fields are important for these rules to work.