Types of Polynomials
Polynomials come in different shapes and sizes, and mathematicians have developed a systematic way to classify them. In Class 9, you learn to classify polynomials using two criteria: the number of terms and the degree (highest power of the variable). Based on terms, a polynomial can be a monomial (1 term), binomial (2 terms), or trinomial (3 terms). Based on degree, it can be a constant (degree 0), linear (degree 1), quadratic (degree 2), cubic (degree 3), or higher. These classifications are not just academic labels — they tell you a great deal about how the polynomial behaves, what its graph looks like, how many zeroes it can have, and what methods you can use to factorise or solve it. For example, knowing a polynomial is a 'quadratic trinomial' immediately tells you it has degree 2, three terms, and can potentially be factorised by splitting the middle term. This topic gives you the vocabulary and framework to describe any polynomial precisely, which is essential as you work through the rest of the Polynomials chapter — from finding zeroes and applying the Remainder Theorem to factorisation and using algebraic identities.
What is Types of Polynomials?
Polynomials are classified in two independent ways:
Classification 1: By Number of Terms
Monomial: A polynomial with exactly one term.
Examples: 5x^3, -7, 2y, sqrt(3)x^2, 0.5.
A monomial has the form ax^n (coefficient times variable raised to a power).
Binomial: A polynomial with exactly two terms.
Examples: x + 5, 3x^2 - 2x, y^4 + 1, 2x^3 - 7x.
The two terms must have different degrees (otherwise they would combine into one term).
Trinomial: A polynomial with exactly three terms.
Examples: x^2 + 3x + 2, 2y^3 - y + 7, a^2 + 2ab + b^2.
Many standard quadratic expressions are trinomials.
Polynomials with four or more terms are simply called polynomials — there is no special name for them.
Classification 2: By Degree
Constant Polynomial (Degree 0): No variable term. Examples: 5, -3, sqrt(2).
Graph: horizontal line. Zero zeroes (for non-zero constants).
Linear Polynomial (Degree 1): Highest power is 1. General form: ax + b, a not 0.
Examples: 3x + 7, -x + 4, (2/3)x.
Graph: straight line. Exactly 1 zero.
Quadratic Polynomial (Degree 2): Highest power is 2. General form: ax^2 + bx + c, a not 0.
Examples: x^2 - 4x + 3, 2x^2 + 1, -x^2 + 5x.
Graph: parabola. At most 2 zeroes.
Cubic Polynomial (Degree 3): Highest power is 3. General form: ax^3 + bx^2 + cx + d, a not 0.
Examples: x^3 - 6x^2 + 11x - 6, 2x^3 + 1.
Graph: S-shaped curve. At most 3 zeroes.
Biquadratic (Quartic) Polynomial (Degree 4): Highest power is 4. General form: ax^4 + bx^3 + cx^2 + dx + e, a not 0.
Examples: x^4 - 5x^2 + 4, x^4 + 1.
Combined classification: You can describe a polynomial by both criteria. For example, x^2 + 5x + 6 is a 'quadratic trinomial' (degree 2, three terms), and 4x^3 is a 'cubic monomial' (degree 3, one term).
Types of Polynomials Formula
General forms for each type (by degree):
Constant: p(x) = a_0, where a_0 is not 0.
Zero of constant polynomial: None (for a_0 not 0).
Linear: p(x) = ax + b, where a is not 0.
Zero: x = -b/a (exactly one).
Quadratic: p(x) = ax^2 + bx + c, where a is not 0.
Zeroes: x = (-b ± sqrt(b^2 - 4ac)) / (2a) (quadratic formula).
Discriminant: D = b^2 - 4ac determines the number of real zeroes:
D > 0: two distinct real zeroes
D = 0: one repeated real zero
D < 0: no real zeroes
Cubic: p(x) = ax^3 + bx^2 + cx + d, where a is not 0.
At least one real zero always exists. Up to 3 real zeroes.
Number of terms — counting rules:
1. Like terms must be combined before counting. 3x^2 + 5x^2 = 8x^2 is a monomial, not a binomial.
2. The constant term (if non-zero) counts as a term.
3. Terms with zero coefficients do not count. In 0x^3 + 2x + 1, there are 2 terms (not 3).
Derivation and Proof
Why these classifications matter — a practical perspective:
The classification of polynomials is not arbitrary. Each type has specific properties and solution methods.
Why degree matters:
The degree determines the maximum number of zeroes and the shape of the graph.
A linear polynomial ax + b = 0 gives one solution. Method: simple algebra (isolate x).
A quadratic ax^2 + bx + c = 0 gives up to two solutions. Methods: factorisation, completing the square, or the quadratic formula.
A cubic polynomial gives up to three solutions. Methods: factor theorem, trial and error for one root, then reduce to quadratic.
As the degree increases, the polynomial becomes more complex, and more sophisticated methods are needed.
Why the number of terms matters:
Monomials are the simplest — they can be factored immediately (just pull out the common factor). Example: 6x^3 = 6 x x x x.
Binomials often fit standard identities: a^2 - b^2 = (a+b)(a-b), a^3 + b^3 = (a+b)(a^2 - ab + b^2), etc.
Trinomials often arise from expanding products of two binomials: (x + p)(x + q) = x^2 + (p+q)x + pq. This is why 'splitting the middle term' works for factorising quadratic trinomials.
Connection between the two classifications:
A quadratic polynomial MUST have at least 1 term (the ax^2 term) and at most 3 terms (ax^2 + bx + c). It can be a monomial (3x^2), binomial (x^2 + 5), or trinomial (x^2 - x + 1). Its degree is always 2 regardless of the number of terms.
Types and Properties
Here is a comprehensive summary of all polynomial types with examples:
By Number of Terms:
Monomial (1 term):
- 7x^4 (quartic monomial)
- -3 (constant monomial)
- 2x (linear monomial)
- x^2 (quadratic monomial)
Binomial (2 terms):
- x + 5 (linear binomial)
- x^2 - 9 (quadratic binomial, also a difference of squares)
- x^3 + 8 (cubic binomial, also a sum of cubes)
- 3x^4 - x (quartic binomial)
Trinomial (3 terms):
- x^2 + 5x + 6 (quadratic trinomial — the most common type in school maths)
- x^3 - 3x + 2 (cubic trinomial)
- 2x^4 + x^2 - 1 (quartic trinomial)
Polynomial with more than 3 terms:
- x^3 + 2x^2 - 5x + 1 (4 terms, cubic)
- x^4 - x^3 + x^2 - x + 1 (5 terms, quartic)
By Degree:
Constant (Degree 0): 5, -sqrt(3), 1/2.
Zero polynomial (p(x) = 0) is constant but degree is undefined.
Linear (Degree 1): 2x + 3, -x, (1/4)x - 7.
Quadratic (Degree 2): x^2 + 3x + 2, -2x^2 + x, 4x^2.
Cubic (Degree 3): x^3 - 1, 2x^3 + 5x^2 - x + 3, -x^3.
Biquadratic/Quartic (Degree 4): x^4, x^4 - 5x^2 + 4, 3x^4 + x.
Combined classification examples:
x^2 + 5x + 6 → Quadratic Trinomial
4x^3 → Cubic Monomial
x + 7 → Linear Binomial
x^2 - 16 → Quadratic Binomial
-9 → Constant Monomial
Solved Examples
Example 1: Example 1: Classifying by number of terms
Problem: Classify as monomial, binomial, or trinomial:
(a) 5x^2 - 3x + 1 (b) 7y (c) x^2 + 9 (d) -3 (e) 2a^3 - a + 5a^2
Solution:
(a) 5x^2 - 3x + 1: Three terms → Trinomial.
(b) 7y: One term → Monomial.
(c) x^2 + 9: Two terms → Binomial.
(d) -3: One term (a constant) → Monomial.
(e) 2a^3 - a + 5a^2 = 2a^3 + 5a^2 - a: Three terms → Trinomial.
Example 2: Example 2: Classifying by degree
Problem: State the type (by degree) of each polynomial:
(a) 4x - 9 (b) x^3 + 2x^2 - x + 3 (c) sqrt(5) (d) -x^2 + 7x (e) x^4 + x
Solution:
(a) Degree 1 → Linear.
(b) Degree 3 → Cubic.
(c) Degree 0 → Constant.
(d) Degree 2 → Quadratic.
(e) Degree 4 → Biquadratic (Quartic).
Example 3: Example 3: Combined classification
Problem: Give the combined classification (degree type + term type) for each:
(a) 3x^2 - 7 (b) -2x^3 (c) x^2 + 4x + 4 (d) 5x - 1
Solution:
(a) Degree 2, 2 terms → Quadratic Binomial.
(b) Degree 3, 1 term → Cubic Monomial.
(c) Degree 2, 3 terms → Quadratic Trinomial.
(d) Degree 1, 2 terms → Linear Binomial.
Example 4: Example 4: Combining like terms before classifying
Problem: Classify 3x^2 + 5x - 2x^2 + 7 - 5x.
Solution:
First, combine like terms:
(3x^2 - 2x^2) + (5x - 5x) + 7 = x^2 + 0 + 7 = x^2 + 7.
After simplification: x^2 + 7 has 2 terms and degree 2.
Classification: Quadratic Binomial.
Lesson: Always simplify by combining like terms before classifying!
Example 5: Example 5: Writing polynomials of given types
Problem: Write one example of each:
(a) A cubic binomial
(b) A linear monomial
(c) A quadratic trinomial with all positive coefficients
(d) A constant polynomial
Solution:
(a) Cubic binomial: x^3 - 27 (degree 3, 2 terms).
(b) Linear monomial: -4x (degree 1, 1 term).
(c) Quadratic trinomial, all positive: 2x^2 + 3x + 1 (degree 2, 3 terms, all coefficients positive).
(d) Constant polynomial: 11 (degree 0).
Example 6: Example 6: Determining zeroes based on type
Problem: How many real zeroes can each type have?
(a) Linear polynomial (b) Quadratic polynomial (c) Constant polynomial (non-zero)
Solution:
(a) Linear (degree 1): Exactly 1 real zero. The graph is a straight line that crosses the x-axis at exactly one point.
(b) Quadratic (degree 2): At most 2 real zeroes. It can have 2 (parabola crosses x-axis twice), 1 (touches x-axis), or 0 (parabola is entirely above or below x-axis).
(c) Constant (non-zero, degree 0): 0 zeroes. A horizontal line (not on the x-axis) never crosses the x-axis.
Example 7: Example 7: Identifying special binomial types
Problem: Identify the algebraic pattern in each binomial:
(a) x^2 - 25 (b) 8x^3 + 1 (c) x^2 + 2x
Solution:
(a) x^2 - 25 = x^2 - 5^2: This is a difference of squares. It can be factored as (x + 5)(x - 5).
(b) 8x^3 + 1 = (2x)^3 + 1^3: This is a sum of cubes. It can be factored as (2x + 1)(4x^2 - 2x + 1).
(c) x^2 + 2x = x(x + 2): This has a common factor. It factors as x(x + 2).
Recognising the type of binomial helps you choose the right factorisation method.
Example 8: Example 8: Why x^2 + 4 and x^2 - 4 are very different
Problem: Both x^2 + 4 and x^2 - 4 are quadratic binomials. Compare their zeroes.
Solution:
x^2 - 4 = 0: x^2 = 4, so x = ±2. Two real zeroes (2 and -2).
x^2 + 4 = 0: x^2 = -4. No real number squared gives a negative result. Zero real zeroes.
Despite being the same type (quadratic binomial), these two polynomials have different numbers of zeroes. The sign before the constant term makes all the difference.
x^2 - 4 is a difference of squares and factors nicely. x^2 + 4 is a sum of squares and has no real factorisation.
Example 9: Example 9: Tricky classification questions
Problem: Classify: (a) (x + 1)^2 (b) (x + 2)(x - 2)
Solution:
(a) (x + 1)^2 = x^2 + 2x + 1: After expanding, this is a quadratic trinomial.
(b) (x + 2)(x - 2) = x^2 - 4: After expanding, this is a quadratic binomial.
Note: Classification is done on the expanded (simplified) form, not the factored form. The factored form tells you about the structure, but the type is determined by the polynomial in standard form.
Example 10: Example 10: All possible types for degree 2
Problem: List all possible types (by number of terms) for a quadratic polynomial. Give an example of each.
Solution:
A quadratic polynomial has degree 2, so it must contain an x^2 term. It can have:
Quadratic Monomial (1 term): 5x^2 (only the x^2 term; b = 0 and c = 0 in ax^2 + bx + c).
Quadratic Binomial (2 terms): x^2 + 3 (missing the x term) or x^2 - 7x (missing the constant term).
Quadratic Trinomial (3 terms): 2x^2 - 5x + 3 (all three terms present).
A quadratic cannot have more than 3 terms because there are only 3 possible distinct powers: x^2, x^1, and x^0.
Real-World Applications
Understanding the types of polynomials helps in many mathematical and real-world contexts:
Factorisation Strategy: Knowing the type tells you which method to use. Quadratic trinomials → try splitting the middle term. Binomials → look for difference of squares, sum/difference of cubes. Polynomials with a common factor → extract it first. This saves time and avoids trial-and-error.
Equation Solving: Linear equations have a direct formula (x = -b/a). Quadratic equations have the quadratic formula. Cubic equations require the factor theorem. Knowing the type tells you which tool to reach for.
Modelling Real-World Situations: Linear polynomials model constant-rate situations (distance = speed x time). Quadratic polynomials model acceleration (projectile motion, area problems). Cubic polynomials model volume problems. Choosing the right polynomial type is the first step in mathematical modelling.
Graphing: The type determines the graph shape. Linear → line, quadratic → parabola, cubic → S-curve. This helps in sketching and interpreting graphs in science, economics, and data analysis.
Communication in Mathematics: When discussing problems with teachers, peers, or in textbooks, using precise terminology ('quadratic trinomial,' 'cubic binomial') ensures clear and unambiguous communication.
Key Points to Remember
- Polynomials are classified by number of terms: monomial (1), binomial (2), trinomial (3).
- Polynomials are classified by degree: constant (0), linear (1), quadratic (2), cubic (3), biquadratic (4).
- A polynomial can be described using both classifications: e.g., 'quadratic trinomial.'
- Always combine like terms before classifying — the simplified form determines the type.
- The general form of a linear polynomial is ax + b (a not 0).
- The general form of a quadratic polynomial is ax^2 + bx + c (a not 0).
- The general form of a cubic polynomial is ax^3 + bx^2 + cx + d (a not 0).
- The type determines factorisation methods and the maximum number of zeroes.
- Recognising binomial patterns (difference of squares, sum/difference of cubes) is key to factorisation.
- The zero polynomial p(x) = 0 has undefined degree and no special classification.
Practice Problems
- Classify each as monomial, binomial, or trinomial: (a) x^3 - x, (b) 4, (c) 2x^2 - 3x + 5, (d) -7x^5, (e) x + sqrt(2).
- State the degree and type (linear/quadratic/cubic) of each: (a) 3x^2 + 1, (b) x^3 - 8, (c) -5x + 2, (d) 12.
- Write one example of each: (a) quadratic monomial, (b) cubic trinomial, (c) linear binomial, (d) constant monomial.
- Simplify and classify: 4x^2 + 3x - 2x^2 + 7 - 3x.
- For each type of polynomial (linear, quadratic, cubic), write the maximum and minimum number of terms it can have.
- Is it possible to have a quadratic polynomial with 4 terms? Why or why not?
- Give an example of a cubic polynomial that is also a binomial. Can you factor it using an algebraic identity?
- Classify (x - 3)^3 by expanding it first. What is its degree and how many terms does it have?
Frequently Asked Questions
Q1. What are the different types of polynomials?
Polynomials are classified in two ways: (1) By number of terms — monomial (1 term like 5x^2), binomial (2 terms like x + 3), trinomial (3 terms like x^2 + 5x + 6). (2) By degree — constant (degree 0 like 7), linear (degree 1 like 2x + 1), quadratic (degree 2 like x^2 - 4), cubic (degree 3 like x^3 + 1). A polynomial can be described using both: x^2 + 3x + 2 is a 'quadratic trinomial.'
Q2. What is a monomial?
A monomial is a polynomial with exactly one term. It is a product of a number (coefficient) and a variable raised to a non-negative integer power. Examples: 5x^3, -7, 2y, sqrt(3)x^2. The single number 5 is also a monomial (with degree 0). Monomials are the building blocks of all polynomials.
Q3. What is the difference between a binomial and a trinomial?
A binomial has exactly 2 terms (like x^2 - 9 or 3x + 1), while a trinomial has exactly 3 terms (like x^2 + 5x + 6 or 2x^3 - x + 7). The count is done after combining any like terms. So 3x + 2x + 1 is not a trinomial — it simplifies to 5x + 1, which is a binomial.
Q4. What is a quadratic polynomial?
A quadratic polynomial has degree 2, meaning the highest power of the variable is 2. Its general form is ax^2 + bx + c where a is not zero. Examples: x^2 + 5x + 6, 3x^2 - 1, -x^2. The graph of a quadratic is a parabola. It can have 0, 1, or 2 real zeroes. The word 'quadratic' comes from 'quadratus' meaning square.
Q5. Can a monomial be a quadratic?
Yes! A quadratic monomial is a single term with degree 2. For example, 5x^2 is a quadratic monomial. It has degree 2 (so it is quadratic) and only one term (so it is a monomial). Similarly, -x^2 and sqrt(7)x^2 are quadratic monomials.
Q6. What is the difference between a linear and a quadratic polynomial?
A linear polynomial has degree 1 (highest power is x^1), and its graph is a straight line. Example: 3x + 2. It has exactly 1 zero. A quadratic polynomial has degree 2 (highest power is x^2), and its graph is a parabola. Example: x^2 - 4x + 3. It can have 0, 1, or 2 zeroes. The key difference is the degree, which changes the shape of the graph and the number of possible solutions.
Q7. How do you classify a polynomial with 4 or more terms?
Polynomials with 4 or more terms do not have a special name based on the number of terms. They are simply called 'polynomials.' For example, x^3 + 2x^2 - 5x + 1 is a cubic polynomial with 4 terms. You still classify it by degree (cubic) but not by a specific term-count name.
Q8. Is 0 a monomial?
This is debatable in mathematics. In most school textbooks (including NCERT), the zero polynomial p(x) = 0 is considered separately. Some texts treat 0 as a monomial with undefined degree. For Class 9 purposes, remember that non-zero constants like 5, -3 are monomials of degree 0, but the zero polynomial is a special case with undefined degree.
Q9. What type of polynomial is (a + b)^2?
Expand it first: (a + b)^2 = a^2 + 2ab + b^2. This has 3 terms and degree 2 (considering a and b as variables, or degree 2 in a single variable if one is treated as a constant). In one variable, it is a quadratic trinomial. Always expand and simplify before classifying.
Q10. Why is it important to know the type of a polynomial?
Knowing the type helps you choose the right approach for solving or factorising. Binomials often fit standard identities (a^2 - b^2, a^3 ± b^3). Quadratic trinomials can be factored by splitting the middle term. The degree tells you the maximum number of zeroes and the shape of the graph. Type classification is your first step in deciding how to work with a polynomial.
Related Topics
- Degree of a Polynomial
- Polynomials in One Variable
- Zeroes of a Polynomial
- Factorisation of Polynomials
- Value of a Polynomial
- Remainder Theorem
- Factor Theorem
- Algebraic Identities (Extended)
- (a + b)³ and (a - b)³ Identities
- a³ + b³ and a³ - b³ Identities
- Zeroes of Quadratic Polynomial
- Relationship Between Zeroes and Coefficients
- Sum and Product of Zeroes
- Graph of Quadratic Polynomial
