Types of Polynomials Class 10: Degree, Linear, Quadratic & Cubic

Quick answer: In CBSE Class 10 Maths, polynomials are classified by their degree — the highest exponent of the variable. The main focus is on three types: the linear polynomial (degree 1, like 2x − 3), the quadratic polynomial (degree 2, like x² + 7x + 10) and the cubic polynomial (degree 3, like x³ − 4x). A non-zero constant such as 7 is also a polynomial, of degree 0.

A polynomial is an algebraic expression in which the variable is raised only to whole-number exponents (0, 1, 2, 3, …), combined using addition, subtraction and multiplication. So 4x + 2, 2y² − 3y + 4 and 5x³ − 4x² + x − 2 are all polynomial expressions, but 1/x, √x and x⁻² are not, because their variable powers are not whole numbers.

This guide explains the definition, how to read the degree of a polynomial, and each of the three types with simple examples — everything you need for the opening of Chapter 2 (Polynomials), Section 2.1.

Note on accuracy: This explanation follows the latest NCERT Class 10 Mathematics terminology for polynomials, degree and standard forms.

What is a polynomial? (Definition)

A polynomial in one variable x is a polynomial expression built from terms of the form (a number) × (a whole-number power of x). Each number is called a coefficient, and a term with no variable is the constant term. Coefficients can be any real numbers — whole numbers, fractions like 1/2, or irrationals like √2 and π. What decides whether something is a polynomial is the variable powers, not the coefficients.

Examples of polynomials: 4x + 2, 2y² − 3y + 4, 5x³ − 4x² + x − 2, πx² + 3.

Not polynomials: 1/x (which is x⁻¹), √x (which is x^½), and 2/x + 3. In every “not a polynomial” case, the variable appears with a negative or fractional exponent, or in a denominator — which is banned.

The one-second test

Look at every term's variable power
           ↓
 Is every power a whole number (0,1,2,3,…)?
      ↓ YES              ↓ NO
 Polynomial ✅      Not a polynomial ❌

A root sign over the variable, a variable in the denominator, or a negative power instantly means “not a polynomial.”

Polynomial or not? — comparison card

Can you identify it?

Which of these are polynomials?

①  4x³ − 2x + 1
②  √x + 2
③  1/x
④  5
⑤  x² + x + 1

Answers: ①, ④ and ⑤ are polynomials. ② has a fractional power (√x) and ③ has a negative power (1/x), so they are not polynomials.

🎬 Watch (30-sec Short, opens in a new tab): Not a polynomial? — spot the odd one out in 5 seconds.

Want a timed set with instant checking? The Topic Practice on examfront turns this into quick reps.

What is the degree of a polynomial?

The degree of a polynomial is the highest exponent of the variable that appears in it.

Degree is the single most useful label for a polynomial, because the three named types in Class 10 are defined purely by degree. A non-zero constant like 7 has degree 0 (think of it as 7x⁰), while the number 0 alone is the zero polynomial, whose degree is not defined.

🎬 Watch (30-sec Short, opens in a new tab): Degree of a polynomial · Degree of 7? — the constant-degree trick everyone misses.

The degree ladder — each rung is one of the polynomial types:

Degree 0  →  Constant   (e.g. 7)
   ↓
Degree 1  →  Linear     (e.g. 2x − 3)
   ↓
Degree 2  →  Quadratic  (e.g. x² + 7x + 10)
   ↓
Degree 3  →  Cubic      (e.g. x³ − 4x)

examfront Insight Students often read the first term’s power instead of the highest power. Always scan every term before deciding the degree — in 5x⁴ − 3x⁷ + 2, the degree is 7, not 4. examfront’s Mistake Identification flags this exact slip when it happens.

🎬 Watch (30-sec Short, opens in a new tab): Degree shortcut! — find the degree of a product without expanding.

The three types of polynomials in Class 10

In CBSE Class 10, the main focus is on linear, quadratic and cubic polynomials. (A non-zero constant is also a polynomial, of degree 0 — it just isn’t one of the three “named” types you’re tested on.)

Linear polynomial (degree 1)

A linear polynomial is a polynomial of degree 1. Its general form is:

ax + b, where a and b are real numbers and a ≠ 0.

• Meaning in words: the variable appears only to the first power.
• Why a ≠ 0: if a were 0 the x-term would vanish and it would drop to degree 0.
• Examples: 2x − 3, 3z + 4, √3·x − 5.
• Not linear: 2x + 5 − x² (has an x² term, so degree 2).

Worked example. Is 7 − 4x a linear polynomial? Highest power of x is 1 (from −4x), and the coefficient −4 ≠ 0, so yes, it is linear.

Quadratic polynomial (degree 2)

A quadratic polynomial is a polynomial of degree 2. The name comes from quadrate, meaning “square.” Its general form is:

ax² + bx + c, where a, b, c are real numbers and a ≠ 0.

• Meaning in words: the highest power of the variable is 2.
• Each variable defined: a = coefficient of x², b = coefficient of x, c = constant term.
• Condition: a ≠ 0 (otherwise it is not quadratic).
• Examples: x² + 7x + 10, 2x² − 3x, y² − 2.

Worked example. Classify 6x² + x. The highest power is 2 and the x² coefficient is 6 ≠ 0, so it is a quadratic polynomial with a = 6, b = 1, c = 0.

Cubic polynomial (degree 3)

A cubic polynomial is a polynomial of degree 3. Its general form is:

ax³ + bx² + cx + d, where a, b, c, d are real numbers and a ≠ 0.

• Meaning in words: the highest power of the variable is 3.
• Examples: x³, 3x³ − 2x² + x − 1, 2 − x³.

Worked example. Classify 2x³ − x + 5. The highest power is 3 with coefficient 2 ≠ 0, so it is a cubic polynomial (here a = 2, b = 0, c = −1, d = 5).

Quick comparison table

Notice the pattern: each type just adds the next lower-degree terms, and the leading coefficient must never be zero. Once you can name the degree, you can name the type instantly.

🎬 Watch (30-sec Short, opens in a new tab): Name the polynomial! — linear, quadratic or cubic in 3 seconds.

For a full mixed set that jumps between degrees, drill the harder cases inside examfront.

Why knowing polynomial types matters

This is the foundation for most of Chapter 2 and several chapters after it. Polynomial classification helps you later in:

• finding zeroes of a polynomial,
• factorisation and using algebraic identities,
• quadratic equations (a quadratic polynomial set equal to zero),
• reading the graphs of polynomials,
• and answering board questions that first ask you to identify the type.

Getting the type right in one glance makes every one of those steps faster and less error-prone.

Key Takeaways

• A polynomial allows only whole-number exponents of the variable; √x, 1/x and negative powers are not polynomials.
• Coefficients can be any real numbers (fractions, √2, π); only the variable powers decide if it’s a polynomial.
• The degree is the highest exponent of the variable, and it is what defines each type.
• Linear = degree 1 (ax + b), quadratic = degree 2 (ax² + bx + c), cubic = degree 3 (ax³ + bx² + cx + d), each with a ≠ 0.
• A non-zero constant (like 7) is a polynomial of degree 0; the number 0 is the zero polynomial with undefined degree.

Quick Facts

• Chapter: 2 — Polynomials (CBSE Class 10 Maths).
• Definition: Polynomial = terms with whole-number exponents of the variable.
• Degree: highest exponent of the variable.
• Linear: ax + b, a ≠ 0, degree 1.
• Quadratic: ax² + bx + c, a ≠ 0, degree 2.
• Cubic: ax³ + bx² + cx + d, a ≠ 0, degree 3.
• Degree of a non-zero constant: 0.

Common Mistakes

1. Calling √x or 1/x a polynomial. They have fractional/negative powers, so they are not polynomials.
2. Reading the first term as the degree. Degree is the highest power anywhere in the expression, not the power of the first term you see.
3. Forgetting the a ≠ 0 condition. In ax² + bx + c, if a = 0 the expression is no longer quadratic.
4. Thinking a constant has no degree. A non-zero constant has degree 0; only the number 0 has undefined degree.
5. Confusing “polynomial” with “equation.” x² + 7x + 10 is a polynomial; setting it equal to 0 makes it an equation.

FAQ

What are the 3 types of polynomials in Class 10? By degree: linear (degree 1, e.g. 2x − 3), quadratic (degree 2, e.g. x² + 7x + 10) and cubic (degree 3, e.g. x³ − 4x). A non-zero constant is also a polynomial, of degree 0.

How do I find the degree of a polynomial? Find the highest exponent of the variable. In 5x³ − 4x² + x − 2, the highest power is 3, so the degree is 3.

Is √x a polynomial? No. √x = x^(1/2), and ½ is not a whole number. Polynomials allow only whole-number exponents.

Can a polynomial have negative powers? No. Terms like x⁻¹ or 1/x have negative powers and are not allowed in a polynomial.

Is √2x a polynomial? If it means √2 · x, yes — √2 is a coefficient and x has power 1, so it’s linear. If it means √(2x) (variable under the root), no.

Is πx² a polynomial? Yes. π is a real coefficient and x² has a whole-number power, so πx² is a quadratic polynomial.

Can coefficients be fractions? Yes. Coefficients can be any real numbers, including fractions, e.g. (1/2)x + 3.

What is the general form of a quadratic polynomial? ax² + bx + c, where a, b, c are real and a ≠ 0.

Is 7 a polynomial, and what about 0? 7 is a polynomial of degree 0. 0 is the zero polynomial — a polynomial whose degree is not defined.

Related Concepts

• Zeroes of a polynomial — once you can name a polynomial, the next step is finding the values that make it zero. See Zeroes of a Polynomial Class 10.
• Polynomials Class 10 (full chapter guide) — the complete Chapter 2 overview.
• Quadratic equations — what happens when a quadratic polynomial is set equal to zero.
• Algebraic identities — tools for factorising quadratic and cubic polynomials.

Learn in this order

✓ Types of Polynomials            (you are here)
        ↓
✓ Zeroes of a Polynomial
        ↓
✓ Relationship Between Zeroes and Coefficients
        ↓
✓ Division Algorithm for Polynomials

Watch & revise (30-sec Shorts)

Quick recap reels — each opens in a new tab so you stay on this page:

• Degree of 7?
• Not a polynomial?
• Name the polynomial!
• Degree of a polynomial
• Degree shortcut!

Ready to practise? Start with Topic Practice on examfront, then take the Polynomials Chapter Quiz to check your recall — its Mistake Identification shows exactly where a slip happened, and Progress Tracking maps your next step in the sequence above.