Math Lesson 11.1.4 - The Names of Polynomials by Degree

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Welcome to our Math lesson on The Names of Polynomials by Degree, this is the fourth lesson of our suite of math lessons covering the topic of The Definition of Monomials and Polynomials, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Names of Polynomials by Degree

Polynomials are assigned names according to the degree they hold. They are:

1. Zero polynomial. This polynomial has all coefficients, constants and variables equal to zero. In other words, it contains only the number zero. The degree of the zero polynomial either is left undefined or is defined to be negative (usually −1 or -∞).
2. Zeroth degree polynomial. This polynomial - also known as non-zero constant polynomial - contains only one non-zero digit. For example, 3, 14, 127, etc., are all degree zero polynomials.
3. First-degree polynomial, also known as linear polynomial. This type of polynomial contains one or more variable(s) raised to the first power, with or without a constant. For example, P(x) = 2x + 3, P(x,y) = 2x - 3y, P(x, y) = x - 4y + 1, etc., are all first degree polynomials, as their terms contain only variables in the first power and numbers.
4. Second-degree polynomial, which has the monomial with the highest degree 2. In the special case, when such polynomials contain a single variable, they are known as quadratic polynomials, as the variable is raised at the second power at maximum. For example, P(x) = xy + 3x + 1 is a second-degree polynomial, as the first term xy contains two variables at the first power, so when we add their indices gives the degree 1 + 1 = 2. However, this is not a quadratic polynomial, as none of the terms is quadratic. On the other hand, P(x) = 5x² + 3x - 1 is both a second-degree and a quadratic polynomial, as the highest power of the single variable x is 2.
5. Third-degree polynomial, which has the monomial with the highest degree 3. In the special case, when such polynomials contain a single variable, they are known as cubic polynomials, as the variable is raised at the third power at maximum. For example, P(x) = xy² + 5xy + 1 is a third-degree polynomial, as the first term xy² contains two variables at the first power and second power respectively, so when we add their indices gives the degree 1 + 2 = 3. However, this is not a cubic polynomial, as none of the terms is cubic. On the other hand, P(x) = 4x³ + 3x² -3x + 1 is both a third-degree and a cubic polynomial, as the highest power of the single variable x is 3.

We can continue with the same logic to name the higher degree polynomials as shown in the table below.

Example 4

Write the names of the following polynomials in terms of their degree.

1. P(x,y) = 4x³ - x² y² + 3xy - y + 1
2. P(x) = x⁵ - 3x² + 1

Solution 4

1. The degree of the polynomial
   P(x,y) = 4x³ - x² y² + 3xy - y + 1
   is obtained by looking at its second term (-x²y²), as when adding the indices yields 2 + 2 = 4 (the highest total power). Therefore, this is a fourth-degree polynomial. However, this polynomial is not quartic, as it does not have only a single variable raised to the fourth power.
2. The term with the highest power in the polynomial
   P(x) = x⁵ - 3x² + 1
   is the first one. It has the variable x raised to the fifth power. Therefore, this is a fifth-degree polynomial. This is also a quintic polynomial, as since it contains a single variable, it represents a special case of fifth-degree polynomials.

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7. Continuing learning polynomials - read our next math tutorial: Operations with Polynomials

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