1. Classifying Polynomials; But Why?
2. What is a Polynomial?
3. Classify by Degree
4. Classify by Number of Terms
5. Examples
Classifying Polynomials; But Why?
Classifying polynomials might at first seem like a pointless task, but it is an important step in understanding the properties of polynomials and how they behave. Here are some reasons why we classify polynomials:
- Understanding Polynomial Behavior: Polynomials of different degrees and types behave differently, especially when we graph them. For example, even-degree polynomials go in the same direction at both ends of their graphs, while odd-degree polynomials go in opposite directions at either end of their graphs. Linear polynomials produce straight lines, while quadratic polynomials give us parabolas.
- Solving Equations: The degree of a polynomial equation often indicates the number of solutions or roots it has, which is a crucial aspect of algebra. For example, a quadratic polynomial equation has at most two real roots. Understanding the properties of the degree of a polynomial helps us choose the appropriate method for solving it.
- Simplifying Calculations: When we classify polynomials, we can use methods tailored to each type to simplify calculations. For example, binomials can be factored using the difference of squares formula, and cubic polynomials can be factored using synthetic division or other methods.
- Teaching Mathematical Concepts: Polynomials and their classification are also excellent teaching tools for abstract mathematical concepts such as variables, coefficients, exponents, and the Distributive Law.
What is a Polynomial?
In mathematics, a polynomial is an expression consisting of variables of varying degrees and coefficients, with the important condition that all the coefficients are real numbers (not allowed; \sqrt{-1}). It involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
An example of a polynomial of a single variable, x, is x^2 - 4x + 7. This is a polynomial of degree 2, or a quadratic polynomial.
The individual terms of a polynomial are composed of a constant (the coefficient of the term), a variable, and an exponent. For example, in the term 4x^3, 4 is the coefficient, x is the variable, and 3 is the exponent. All the coefficients in a polynomial must be real numbers.
Polynomials can be classified by their degree and by the number of terms they are composed of. The degree of a polynomial is the largest exponent on the variable in the polynomial, while the number of terms is simply the number of separate terms, which are separated by either addition or subtraction.
Polynomials can have one variable or multiple variables. For example, x^2 + 2x + 1 is a polynomial in one variable (x), and x^2y + 3xy^2 - 4x^2y^3 + 7 is a polynomial in two variables (x and y).
More Examples;
\large{ x, 3x^{3}\text{, }\frac{7}{5}x^{31}\text{, }\pi x \text{, }32x^{3}y^{2}z}
More Non-Examples;
\large{ \sqrt{x}\text{, }\frac{1}{x}\text{, }x^{-2}\text{, }\sqrt{-1}x\text{, }x^{\frac{2}{3}} }
Classifying Polynomials
Classifying polynomials can be a breeze when we understand what we’re looking at. When we classify polynomials, we are identifying the polynomials by their degree and the number of terms they have.
Classify by Degree
Let’s start with classifying polynomials by degree. The degree of a polynomial is simply the highest exponent in the polynomial. Here’s a quick rundown of the names we give to polynomials based on their degree, with examples for each.
Typically, you will only be expected to memorize the names up to cubic or quartic. For the higher degree polynomials we usually just refer to them by their ordinal number e.g. quintic polynomial = fifth degree polynomial or septic polynomial = 7th degree polynomial. *Note that these expressions are placed in parenthesis only for ease of reading.
Constant: A polynomial of degree 0.
(\pi), (-3x^{0}), (100).
Linear: A polynomial of degree 1.
(2x+1), (-3x+5), (x^{1}-10).
Quadratic: A polynomial of degree 2.
(x^2+2x+1), (3x^2-5x+7), (x^2+10).
Cubic: A polynomial of degree 3.
(x^3+2x^2-x+1), (-x^3+5x^2-4x+2), (x^3+3)
Quartic: A polynomial of degree 4.
(x^4+2x^3-x^2+x+1), (2x^4-3x^2+4), (-x^4+5)
Quintic: A polynomial of degree 5.
(x^5-3x^4+2x^3-x^2+1), (2x^5+4x^3-3x^2+2x-1), (x^5)
Sextic: A polynomial of degree 6.
(x^6+2x^5-x^4+x^3-x^2+x-1), (2x^6-3x^5-2x^3+3x^2-1), (x^6)
Septic: A polynomial of degree 7.
(x^7-x^6+x^5-x^4+x-1), (2x^7+3x^6-4x^5-3x^3+4x^2-x+1), (x^7)
Octic: A polynomial of degree 8.
(x^8+1), (2x^8+3x^7-3x^4+2x^3-x+1), (x^8)
Nonic: A polynomial of degree 9.
(x^9-2x^8+3x^7-2x^2+x-1), (2x^9+3x^8+5x^6-4x^5+3x^4-2x^3+1), (x^9)
Decic: A polynomial of degree 10.
(x^{10}-2x^9+5x^6+3x^4-2x^3+x^2-x+1), (2x^{10}+5x^7+3x^5), (x^{10})
Classify by Number of Terms
Now let’s identify polynomials by the number of terms they have. When a polynomial consists of only a single term we call that a monomial. We can string multiple monomials together with either addition or subtraction; +/- . As we increase the number of terms in the expression, we use different names to refer to the polynomial.
- Monomial: A polynomial with just one term.
(x^3), (5x^2), (7x) -
Binomial: A polynomial with two terms.
(x^2+1), (2x^3-5x^2), (3x+7) - Trinomial: A polynomial with three terms.
(x^2+2x+1), (x^3-4x^2+3), (5x^2+2x-1)
- Quadrinomial: A polynomial with four terms.
(2x^{10}+5x^7+3x^5-10)
- Quintrinomial: A polynomial with five terms.
(2x^5+4x^3-3x^2+2x-1)
- Polynomial: This is a catch-all term that we use to describe any algebraic expression with one or more terms. And typically we use this term for polynomials of four or greater terms. I’ve included here the names for four and five term polynomials for those curious readers.
Classifying Polynomials by Degree and Number of Terms Together
Now we put it all together. When faced with a polynomial, we can often gain a lot of information by classifying it based on both its degree and its number of terms. When we use both classifications together, we say the degree classification followed by the number of terms classification. Here are some examples to illustrate this process:
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- 7x^4: This polynomial has one term, which is of the 4th degree. So, it’s a quartic monomial.
- x^3 - 5x^2: This polynomial has two terms, the highest degree among which is 3. Thus, it’s a cubic binomial.
- 3x^5 + 2x - 1: This polynomial has three terms, and the highest degree is 5. So, it’s a quintic trinomial.
- 2x^6 - x^4 + x^2 - x: This polynomial has four terms, the highest degree among which is 6. Therefore, it’s a sextic polynomial.
- x^{10} + x^7 - x^5 + 3x^3 - 2x + 1: This polynomial has six terms, and the highest degree is 10. Hence, it’s a decic polynomial.
- 7x^4: This polynomial has one term, which is of the 4th degree. So, it’s a quartic monomial.
Remember that when you’re classifying a polynomial, you always refer to the highest degree of all the terms for the polynomial’s degree, and the total number of terms for the polynomial’s term type. With practice, you’ll be able to classify polynomials more swiftly and accurately. Keep up the good work, algebra wizards!
