Polynomials are classified either based on their degree or by the number of terms.
Based on Degree
The degree of the polynomial is determined by the highest power of the variable in the leading term. Based on degree, polynomials are classified as:
Types
Function
Example
1. Constant
f(x) = a or f(x) = ax0
f(x) = 7 or 7x0
2. Linear
f(x) = ax + b
f(x) = 2x + 7
3. Quadratic
f(x) = ax2 + bx + c
f(x) = x2 + 2x + 7
4. Cubic
f(x) = ax3 + bx2 + cx + d
f(x) = 3x3 + x2 + 2x + 7
5. Quartic
f(x) = ax4 + bx3 + cx2 + dx + e
f(x) = x4 – 4x3 + 3x2 + 2x – 8
6. Quintic
f(x) = ax5 + bx4 + cx3 + dx2 + ex + f
f(x) = -x5 + x4 – 4x3 + 3x2 + 2x – 8
Note: A function of the form f(x) = 0 (here, ∀ an = 0 and n ∈ ℕ) is called a zero polynomial.
Based on Terms
Polynomials can also be classified by the number of unlike terms (terms with different variables or exponents):
Types
General Form
Example
1. Monomial (single term)
f(x) = axn; here, a ≠ 0
f(x) = 2x
2. Binomial (two unlike terms)
axm + bxn; here, a and b are constants, and m and n are non-negative integers
f(x) = 2x + 7
3. Trinomial (three unlike terms)
axm + bxn + cxp; here, a, b, and c are constants, and m, n, and p are non-negative integers
f(x) = x2 + 2x + 7
In addition, polynomials can have more than three terms. For example, polynomials with four and five unlike terms are called four-term and five-term polynomials, respectively.
Special Types
Here are some special types of polynomials:
Monic Polynomial: It has a leading coefficient of 1. An example is f(x) = x2 – 5x + 4
Irreducible Polynomial: It cannot be factored into polynomials of lower degrees over its given field. An example is f(x, y) = x + y
Homogeneous Polynomial: It has all terms with the same total degree. An example is f(x, y) = x2 + 2xy + y2
Solved Examples
Identify the degree and classify the following polynomials as constant, linear, quadratic, cubic, quartic, or quintic. a) f(x) = 5 b) g(x) = -x3 + 5x2 – x + 7 c) h(x) = 2x2 + 3x + 1 d) k(x) = 4x + 3 e) m(x) = 4x5 – 3x4 + 2x3 – x + 10 f) p(x) = 3x4 – x3 + 6x – 8
Solution:
a) Here, f(x) = 5 f(x) has a degree of 0 Thus, it is a constant polynomial. b) Here, g(x) = -x3 + 5x2 – x + 7 g(x) has a degree of 3 Thus, it is a cubic polynomial. c) Here, h(x) = 2x2 + 3x + 1 h(x) has a degree of 2 Thus, it is a quadratic polynomial. d) Here, k(x) = 4x + 3 k(x) has a degree of 1 Thus, it is a linear polynomial. e) Here, m(x) = 4x5 – 3x4 + 2x3 – x + 10 m(x) has a degree of 5 Thus, it is a quintic polynomial. f) Here, p(x) = 3x4 – x3 + 6x – 8 p(x) has a degree of 4 Thus, it is a quartic polynomial.
Classify by ‘Number of Terms’ and determine if each polynomial is a monomial, binomial, trinomial, or has more than three terms. a) f(x) = -7x4 b) g(x) = 3x3 – 2x c) h(x) = x5 – x + 10 d) k(x) = x3 + y3 e) m(x) = x2 f) n(x) = x2 + 6
Solution:
a) Here, f(x) = -7x4 f(x) has a single term. Thus, it is a monomial. b) Here, g(x) = 3x3 – 2x g(x) has exactly 2 distinct terms. Thus, it is a binomial. c) Here, h(x) = x5 – x + 10 h(x) has 3 distinct terms. Thus, it is a trinomial. d) Here, k(x) = x3 + y3 k(x) has 2 distinct terms. Thus, it is a binomial. e) Here, m(x) = x2 m(x) has a single term. Thus, it is a monomial. f) Here, n(x) = x2 + 6 n(x) has 2 distinct terms. Thus, it is a binomial.
Identify each polynomial to determine whether it is monic, irreducible, or homogeneous. a) f(x) = x3 + 2x2 + x b) g(x) = x2 – 4x + 4 c) h(x, y) = x + y d) k(x, y) = 3x2 + 6xy + 3y2
Solution:
a) Here, f(x) = x3 + 2x2 + x The leading coefficient of f(x) is 1 Thus, it is a monic polynomial. b) Here, g(x) = x2 – 4x + 4 The leading coefficient of g(x) is 1 Thus, it is a monic polynomial. c) Here, h(x, y) = x + y h(x, y) cannot be factored into the lower-degree polynomials. Thus, it is an irreducible polynomial. d) Here, k(x, y) = 3x2 + 6xy + 3y2 k(x, y) has all terms of degree 2 Thus, it is a homogeneous polynomial.
