Polynomials is one of the most important chapters in Class 10 Maths because it connects algebraic expressions, degree, zeros, graphs, and formula-based reasoning in one place. This chapter may look short, but it is highly important for board exams because questions can come from basic definitions, degree of a polynomial, graph-based zeros, relation between zeros and coefficients, and quadratic polynomial formulas.
Many students feel comfortable with the basic examples but get confused when a question asks whether an expression is really a polynomial, how to identify the degree, how many zeros are possible, or how to use α and β formulas. That is why a good quick revision guide is very useful for this chapter.
At Deeksha Vedantu, we always encourage students to revise Polynomials conceptually. Once students understand what a polynomial really is and how zeros and coefficients are connected, the chapter becomes much easier to revise and score from.
Chapter Overview at a Glance
This quick table helps students revise the whole chapter faster.
Quick Concept Table
| Topic | Key idea |
| Polynomial | An algebraic expression with non-negative whole number exponents |
| Degree | Highest power of the variable |
| Zero of a polynomial | A value that makes the polynomial equal to 0 |
| Graphical zero | Point where the graph cuts or touches the x-axis |
| Linear polynomial | Degree 1 |
| Quadratic polynomial | Degree 2 |
| Cubic polynomial | Degree 3 |
| For ax² + bx + c | α + β = -b/a and αβ = c/a |
What Is a Polynomial
A polynomial is an algebraic expression made up of variables, coefficients, and exponents, where the exponents of the variables are non-negative whole numbers.
Standard Idea
A polynomial may contain:
- variables such as x or y
- coefficients such as 2, 5, or -3
- constants such as 7 or 10
Important Rule
For an expression to be a polynomial, the exponent of the variable must be:
- non-negative
- a whole number
This means:
- negative powers are not allowed
- fractional powers are not allowed
- variables in the denominator are not allowed
- roots involving variables are not allowed in polynomial form
Polynomial and Non-Polynomial Examples
Students should be able to identify quickly which expressions are polynomials and which are not.
Examples of Polynomials
| Expression | Is it a polynomial? | Why? |
| 3x² – 5x + 7 | Yes | Powers are 2, 1, and 0 |
| y³ + 2y² – 4 | Yes | All powers are whole and non-negative |
| 5x² + 3x + 1 | Yes | Standard quadratic polynomial |
| 7 | Yes | Constant polynomial |
Examples That Are Not Polynomials
| Expression | Is it a polynomial? | Why not? |
| 2/x | No | Variable is in the denominator |
| √x + 5 | No | Variable has fractional power |
| 5/x² | No | Variable is in the denominator |
| x¹ᐟ² | No | Exponent is not a whole number |
| x⁻² | No | Exponent is negative |
Variable, Coefficient, and Constant
Students should be clear with these basic terms.
Key Terms Table
| Term | Meaning | Example from 3x² – 5x + 7 |
| Variable | Symbol whose value can change | x |
| Coefficient | Number attached to the variable term | 3 and -5 |
| Constant | Term without a variable | 7 |
Degree of a Polynomial
The degree of a polynomial is the highest exponent of the variable in the polynomial. This is one of the most frequently asked direct concepts in board exams.
Types of Polynomials by Degree
| Type | Degree | Example |
| Constant polynomial | 0 | 7 |
| Linear polynomial | 1 | 4x – 1 |
| Quadratic polynomial | 2 | x² – 3x + 2 |
| Cubic polynomial | 3 | 2x³ + x² – 5 |
Important Rule
When more than one variable term is present, always look at the highest power.
For example, in x² + x + 5, the degree is 2 because the highest exponent is 2.
Types of Polynomials by Number of Terms
Students should also revise the classification by number of terms.
Types by Number of Terms Table
| Type | Number of terms | Example |
| Monomial | 1 | 5x² |
| Binomial | 2 | x + 3 |
| Trinomial | 3 | x² – 5x + 6 |
Board-Style Example: Find the Degree
Find the degree of the polynomial:
5x² – 4x + 3
Solution
The variable x appears with powers 2 and 1.
The highest power is 2.
Answer
Degree = 2
Zeros of a Polynomial
A zero of a polynomial is a value of the variable that makes the polynomial equal to zero.
Simple Meaning
If p(x) is a polynomial, then x = a is called a zero if:
p(a) = 0
Examples of Zeros
| Polynomial | Zero condition | Zero |
| 2x – 3 | 2x – 3 = 0 | x = 3/2 |
| x – 5 | x – 5 = 0 | x = 5 |
| x² – 4 | x² – 4 = 0 | x = 2 and x = -2 |
Graphical Representation of Zeros
This is one of the easiest and most important concepts in the chapter.
Rule
The number of points where the graph cuts or touches the x-axis gives the number of zeros of the polynomial.
Important Note
If the polynomial is in x, then count intersection or touching points on the x-axis only.
Number of Zeros by Degree
| Type of polynomial | Maximum number of zeros |
| Linear polynomial | 1 |
| Quadratic polynomial | 2 |
| Cubic polynomial | 3 |
Graph-Based Understanding of Polynomials
The shape of the graph also helps students understand the number of zeros.
Graph Summary Table
| Polynomial type | Graph idea | Zeros visible on graph |
| Linear polynomial | Straight line | Usually 1 zero |
| Quadratic polynomial | Parabola | 0, 1, or 2 real zeros |
| Cubic polynomial | Curve with turning behaviour | Up to 3 zeros |
Linear, Quadratic, and Cubic Polynomials
Students should revise the three most important types carefully.
Linear Polynomial
A linear polynomial has degree 1.
General Form
ax + b, where a ≠ 0
Key Facts
| Feature | Value |
| Degree | 1 |
| Graph | Straight line |
| Number of zeros | 1 |
Quadratic Polynomial
A quadratic polynomial has degree 2.
General Form
ax² + bx + c, where a ≠ 0
Key Facts
| Feature | Value |
| Degree | 2 |
| Graph | Parabola |
| Number of real zeros | 0, 1, or 2 |
Cubic Polynomial
A cubic polynomial has degree 3.
General Form
ax³ + bx² + cx + d, where a ≠ 0
Key Facts
| Feature | Value |
| Degree | 3 |
| Graph | Cubic curve |
| Maximum number of zeros | 3 |
Relation Between Zeros and Coefficients of a Quadratic Polynomial
This is one of the most important parts of Polynomials for board exams.
If α and β are the zeros of the quadratic polynomial:
ax² + bx + c
then the following relations hold.
Main Formula Table
| Concept | Formula |
| Sum of zeros | α + β = -b/a |
| Product of zeros | αβ = c/a |
Students should memorise both formulas clearly.
Why These Formulas Are Important
These formulas help in:
- finding the sum of roots quickly
- finding the product of roots quickly
- forming a quadratic polynomial from given zeros
- solving board-based identity questions
Forming a Quadratic Polynomial When Zeros Are Given
If α and β are the zeros, then the quadratic polynomial with leading coefficient 1 is:
x² – (α + β)x + αβ
More generally, any non-zero constant multiple of this expression also represents a polynomial with the same zeros.
Example: Form Polynomial from Given Zeros
If the zeros are 2 and 3, form the quadratic polynomial.
Step 1: Find Sum and Product
- α + β = 2 + 3 = 5
- αβ = 2 × 3 = 6
Step 2: Use the Formula
x² – (α + β)x + αβ
= x² – 5x + 6
Answer
The required quadratic polynomial is:
x² – 5x + 6
Discriminant and Nature of Zeros
For a quadratic polynomial or quadratic equation:
ax² + bx + c
the discriminant is:
D = b² – 4ac
This helps us decide the nature of zeros.
Case 1: D < 0
There are no real zeros.
Case 2: D = 0
There are two real and equal zeros.
Case 3: D > 0
There are two distinct real zeros.
Discriminant Summary Table
| Condition | Nature of zeros |
| D < 0 | No real zeros |
| D = 0 | Two real and equal zeros |
| D > 0 | Two distinct real zeros |
This is very useful in conceptual and one-mark questions.
Special Formulas Using α and β
These identities are useful for quick board revision. They become easy when students connect them with α + β and αβ.
Identity Formula Table
| Formula number | Identity |
| Formula 1 | α² + β² = (α + β)² – 2αβ |
| Formula 2 | 1/α + 1/β = (α + β)/(αβ) |
| Formula 3 | α³ + β³ = (α + β)³ – 3αβ(α + β) |
| Formula 4 | (α – β)² = (α + β)² – 4αβ |
Board-Style Example: Find α² + β²
For the quadratic polynomial:
x² – 5x + 3 = 0
find α² + β².
Step 1: Use Relation Between Zeros and Coefficients
For x² – 5x + 3:
- α + β = 5
- αβ = 3
Step 2: Use the Formula
α² + β² = (α + β)² – 2αβ
= 5² – 2 × 3
= 25 – 6
= 19
Answer
α² + β² = 19
Board-Style Example: Find α⁴β³ + α³β⁴
For the polynomial:
y² – 5y + 3 = 0
find:
α⁴β³ + α³β⁴
Step 1: Factor the Expression
Take common:
α³β³(α + β)
Step 2: Use Sum and Product of Zeros
From the polynomial:
- α + β = 5
- αβ = 3
So:
α³β³(α + β)
= (αβ)³(α + β)
= 3³ × 5
= 27 × 5
= 135
Answer
The value is 135.
Common Board Questions from Polynomials
This chapter often produces questions from the following areas.
Board Question Types Table
| Area | Common question types |
| Direct definition questions | what is a polynomial, which expression is a polynomial, identify variable or constant |
| Degree-based questions | find the degree, identify type by degree |
| Zero-based questions | find zeros, count zeros from graph, relate zeros to x-axis intersections |
| Formula-based questions | relation between zeros and coefficients, form quadratic polynomial, evaluate identities using α and β |
Quick Revision Guide for Polynomials
This section is useful before exams.
Core Rules Table
| Rule | What to remember |
| Polynomial rule | Exponents must be non-negative whole numbers |
| Degree rule | Degree is the highest exponent |
| Zero rule | A zero makes the polynomial equal to 0 |
| Graph rule | Number of x-axis intersections or touching points gives the number of zeros |
| Sum of zeros | For ax² + bx + c, α + β = -b/a |
| Product of zeros | For ax² + bx + c, αβ = c/a |
Common Mistakes Students Make in Polynomials
Common Mistakes Table
| Mistake | Correct idea |
| Treating √x as a polynomial | It is not a polynomial because the exponent is not a whole number |
| Ignoring negative exponents | If the variable has a negative exponent, it is not a polynomial |
| Confusing degree with number of terms | Degree depends on the highest exponent, not the number of terms |
| Counting zeros incorrectly from graph | Count only the x-axis intersections or touching points |
| Forgetting signs in zero formulas | The formulas are α + β = -b/a and αβ = c/a |
Best Study Strategy for Polynomials
Students can revise this chapter more effectively by following a simple sequence.
Step 1: Revise the Definition Properly
Make sure you know what is and is not a polynomial.
Step 2: Practise Degree Questions
These are easy scoring questions.
Step 3: Revise Graph Meaning of Zeros
This helps in MCQs and concept-based questions.
Step 4: Memorise Sum and Product of Zeros Formulas
These are essential for board exams.
Step 5: Solve Identity-Based Questions
Questions involving α and β become easier when students use formulas instead of expanding blindly.
Practice Questions for Students
Important Practice Questions
- Which of the following is a polynomial?
- Find the degree of 7x³ – 4x + 9.
- How many zeros can a quadratic polynomial have?
- Form a quadratic polynomial whose zeros are 4 and 5.
- For x² – 6x + 8, find α + β and αβ.
- If α and β are the zeros of x² – 4x + 2, find α² + β².
FAQs
Q1. What is a polynomial in Class 10 Maths?
A polynomial is an algebraic expression in which the exponents of the variables are non-negative whole numbers.
Q2. Is √x a polynomial?
No. √x is not a polynomial because its exponent becomes 1/2, which is not a whole number.
Q3. What is the degree of a polynomial?
The degree of a polynomial is the highest power of the variable in that polynomial.
Q4. What are zeros of a polynomial?
Zeros are the values of the variable that make the polynomial equal to zero.
Q5. How do we find the number of zeros from a graph?
Count the number of points where the graph intersects or touches the x-axis.
Q6. What is the relation between zeros and coefficients of a quadratic polynomial?
If α and β are the zeros of ax² + bx + c, then α + β = -b/a and αβ = c/a.
Q7. How do I form a quadratic polynomial if zeros are given?
Use the formula x² – (sum of zeros)x + product of zeros.
Q8. Why is Polynomials important for CBSE board exams?
Polynomials is important because it includes direct definitions, graph-based questions, degree-based questions, and formula-based questions that are commonly asked in board exams.
Conclusion
Polynomials is a compact but highly important Class 10 Maths chapter. It covers definitions, degree, zeros, graphical meaning, and the relationship between zeros and coefficients in a very structured way. Students who revise the chapter carefully can score well because many questions are concept-based and formula-driven rather than lengthy.
The best way to revise Polynomials is to focus first on what makes an expression a polynomial, then move to degree, zeros, graphs, and quadratic formulas. At Deeksha Vedantu, we always encourage students to revise formulas with meaning and practise standard board-style questions so that the chapter feels simple, clear, and scoring.
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