Classifying Polynomials

A polynomial is a mathematical expression consisting of one or more algebraic terms. Each of which has a constant multiplied by one or more variables raised to a non-negative integral power (such as a + bx + cx2). In a polynomial, terms are separated by the addition or the subtraction sign. Multiplication and division operators can not be used to create more terms in a polynomial. For example, 13zy is 13 × z × y, which is a monomial, whereas 13z+y is a binomial.

Classification of polynomials

We can classify polynomials in two ways:

• Based on the number of its terms: Classification based on the number of terms follows a general pattern of prefixing the words ‘mono’, ‘bi’ and ‘tri’ to ‘nomial’. Mono refers to one, bi refers to two, and tri refers to three. The most frequently referred polynomials are monomial, binomial, and trinomial.

• Based on its degree: Classification based on its degree is done by identifying the highest power of the variable that occurs in the polynomial.

Classification based on the number of terms

1. Monomial

A monomial is a polynomial containing one non-zero term.

Illustration

8x5

There is only one non-zero term in this polynomial, i.e., 8×5. Therefore, it is an example of a monomial.

-133y2

There is only one non-zero term in this polynomial, i.e., 133y-2. Therefore, it is an example of a monomial.

2. Binomial

A binomial is a polynomial containing two non-zero terms.

Illustration

17x4−3x3

There are two non-zero terms in this polynomial, i.e., 17x4 and 3x3. Therefore, it is an example of a binomial.

34x9−13x4

There are two non-zero terms in this polynomial, i.e., 34x9 and 13x4. Therefore, it is an example of a binomial.

3. Trinomial

A trinomial is a polynomial containing three non-zero terms.

Illustration

5x3+13x−28

Here, we have three non-zero terms, i.e., 5x3, 13x, and 28. Therefore, it is an example of a trinomial.

65x2+1x−8

Here, we have three non-zero terms, i.e., 65x2, 1x, and 8. Therefore, it is an example of a trinomial.

What is the degree of a polynomial?

If a polynomial involves one variable, the highest power of the variable is called the degree of the polynomial.

Illustration

13x2 + 5x3+17x+83

It is one variable polynomial, i.e., x.

The powers of the x variable are 2, 3, and 1. Therefore, the highest power is 3.

Thus, the degree of a polynomial is 3.

x5+x3+x2+3

It is one variable polynomial, i.e., x.

The powers of the x variable are 5, 3, and 2. Therefore, the highest power is 5.

Thus, the degree of a polynomial is 5.

Classification based on the polynomial’s degree

1. Constant polynomial

It is a polynomial of degree 0.

Illustration

32x0-9x0

We know that x0 is equal to 1, i.e., x0=1.

Thus, the degree of this polynomial is 0. Thus, it is a constant polynomial.

8x0-x0

We know that x0 is equal to 1, i.e., x0=1.

Thus, the degree of this polynomial is 0. Thus, it is a constant polynomial.

2. Linear polynomial

It is a polynomial of degree 1. A linear polynomial has at most two terms in one variable.

General form: ax+b.

Illustration

x1+8

Here, the degree of this polynomial is 1. Thus, it is a linear polynomial.

70x1+3

Here, the degree of this polynomial is 1. Thus, it is a linear polynomial.

3. Quadratic polynomial

It is a polynomial of degree 2. A quadratic polynomial has at most three terms in one variable. 

General form: ax2+bx+c.

Illustration

15x2+12x−7

Here, the degree of this polynomial is 3. Thus, it is a quadratic polynomial.

x2+x−7

Here, the degree of this polynomial is 3. Thus, it is a quadratic polynomial.

4. Cubic polynomial

It is a polynomial of degree 3. A cubic polynomial has at most four terms in one variable.

General form: ax3+bx2+cx+d.

Illustration

12x3+3x2−15x+7

Here, the degree of this polynomial is 3. Thus, it is a cubic polynomial.

x3+x2−x+17

Here, the degree of this polynomial is 3. Thus, it is a cubic polynomial.

5. Quartic polynomial

It is a polynomial of degree 4. A quartic polynomial has at most five terms in one variable.

General form: ax4+bx3+cx2+dx+c

Illustration

32x4− 23x2+ 5x3+ 7x − 4

Here, the degree of this polynomial is 4. Thus, it is a biquadratic polynomial.

x4−x2+x3+x−42

Here, the degree of this polynomial is 4. Thus, it is a biquadratic polynomial.

Conclusion

Polynomials can be classified based on two methods and firstly based on the number of its terms. Secondly, based on its degree.

Classification based on the number of terms is as follows:

• A monomial has just one term. For example, 14x6.

• To be noted, a term contains both the variable(s) and its coefficient. Which, when combined, forms one term.

• A binomial has two terms. For example, 15x2 -4x

• A trinomial has three terms. For example: 13z2+15z-2

• Polynomials with four or more than four terms are called polynomials. For example: 12y5+ 7y3– 51y2+19y-2

Classification based on degree is as follows:

• A constant polynomial is a polynomial of degree 0. For example, x0-1

• A linear polynomial is a polynomial of degree 1. For example, x1-1

• A quadratic polynomial is a polynomial of degree 2. For example, x2-1

• A cubic polynomial is a polynomial of degree 3. For example, x3-1

A biquadratic polynomial is a polynomial of degree 4. For example, x4-1