Degree of a polynomial

In mathematics, the degree of a polynomial tells us how complex the polynomial is. It is the highest power or exponent we can find in any term of the polynomial. Each term in a polynomial, called a monomial, has its own degree, which is the sum of the exponents of all the variables in that term. 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 its behavior, especially for large values of the variable. For a polynomial with just one variable, like ( x^4 + 3x^2 + 1 ), the degree is simply the largest exponent, which is 4 in this case. This degree also tells us the maximum number of times a polynomial can cross the x-axis, which is very useful in solving equations and graphing.

Sometimes, polynomials are not written in their simplest form. For example, the expression ( (x + 1)^2 - (x - 1)^2 ) looks complicated, but by expanding it, we find it simplifies to ( 4x ), which has a degree of 1. This shows that even when polynomials look complex, their degree can still be found by simplifying them. Understanding the degree of a polynomial is a key step in many areas of math, from algebra to calculus.

Names of polynomials by degree

Polynomials are named based on their degree, which is the highest exponent in the polynomial. For example, a polynomial with degree 2 is called quadratic, and 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 quickly describe how complicated a polynomial is. For higher degrees, the names follow Latin ordinal numbers and end in -ic.

Examples

Polynomials are expressions made from variables and numbers using only 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 simply the number 0, is a special case in mathematics. Because it has no terms with numbers other than zero, we usually say it has no degree. However, to make rules about adding and multiplying polynomials work smoothly, some people choose to say its degree is negative infinity.

This choice 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

There are ways to find the degree of a polynomial by looking at how the function behaves when the input values get very large. One method uses a special kind of math called asymptotic analysis, which looks at how the function grows compared to a simple power of the input.

This method can even help us understand the "degree" of some functions that aren’t polynomials, like the multiplicative inverse (1/x), whose degree is -1, or the square root function, whose degree is 1/2. For the logarithm function, the degree is 0, and for the exponential function, the degree is considered to be 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 mathematical structure called a field, the degree of a polynomial behaves in a very useful way. If you multiply two polynomials, the degree of the result is simply the sum of their degrees. This makes the degree function act like a "norm," which helps mathematicians study these polynomials more easily.

However, this nice behavior can fail if we work with other types of structures that aren't fields. For example, in a certain system where numbers are only considered modulo 4, multiplying two degree-1 polynomials can sometimes give a result of degree 0, which breaks the usual rule. This shows why fields are important for these properties to hold.