Introduction

Find the most reliable NCERT Solutions Class 10 Maths Chapter 2 Polynomials at Vidyayan Academy Updated for Session 2025-26. These step-by-step NCERT Solutions make it easy to understand important concepts of Class 10 Maths Chapter 2 Polynomials like degree of polynomials, factorization, remainder theorem, and division algorithm. With clear explanations, students can prepare better and strengthen their basics.

Our NCERT Solutions Class 10 Maths Chapter 2 Polynomials cover NCERT Solution of Exercise 2.1 and Exercise 2.2, and other practice sets in a simple format. Designed to match the latest NCERT curriculum for session 2025-26, these NCERT Solutions help students in revision, homework, and exam preparation. Download free PDFs and start your journey towards scoring full marks in Class 10 Maths.

At Vidyayan Academy, we make this journey simple with clear explanations, solved examples, and 10th Maths NCERT Chapter 2 Solutions. Let’s explore this chapter step by step.

NCERT Solution for Class 10 Maths Chapter 2 Polynomials

Class 10 Maths Exercise 2.1 PDF Download

At Vidyayan Academy, we provide a free PDF for Class 10 Maths Exercise 2.1 of Class 10 Maths Chapter 2 Polynomials chapter that helps students understand the basics of Polynomials step by step. With clear explanations and easy-to-follow methods, this PDF is perfect for practicing offline and revising before exams.

Class 10 Maths Exercise 2.2 PDF Download

For deeper practice, you can also download the Class 10 Maths Exercise 2.2 PDF of Class 10 Maths Chapter 2 Polynomials chpater from Vidyayan Academy. This exercise focuses on applying concepts in a practical way, helping you strengthen your problem-solving skills. Keep it handy for daily revision and effective exam preparation.

Chapter 1 : Real Number Solution

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NCERT Solution for Class 10 Maths Chapter 2 Polynomials all Exercises.

NCERT Solution of Class 10 Maths Exercise 2.1

1.The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Solution:

(i) The number of zeroes is 0 as the graph does not cut the x-axis at any point.

(ii) The number of zeroes is 1 as the graph intersects the x-axis at only 1 point.

(iii) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

(iv) The number of zeroes is 2 as the graph intersects the x-axis at 2 points.

(v) The number of zeroes is 4 as the graph intersects the x-axis at 4 points.

(vi) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.                                    

NCERT Solution of Class 10 Maths Exercise 2.2

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i). x² – 2x – 8

Solution:

= x² – 4x + 2x – 8

= x(x – 4) + 2(x – 4)

= (x + 2)(x – 4)

The value of x² – 2x – 8 is zero if x + 2 = 0 or x – 4 = 0

=> x = -2 or x = 4

Therefore, the zeroes of x² – 2x – 8 are -2 and 4.

Now, Sum of zeroes

= -2 + 4 = 2

= -(-2)/1 = -(Coefficient of x)/Coefficient of x²

Product of zeroes = (-2) x 4 = -8

= -8/1 = Constant term/Coefficient of x².

(ii). 4s² – 4s + 1

Solution:

= 4s² – 2s – 2s + 1

= 2s(2s – 1) – 1(2s – 1)

= (2s – 1)(2s – 1)

The value of 4s² – 4s + 1 is zero if 2s – 1 = 0

=> s = 1/2

Therefore, the zeroes of 4s² – 4s + 1 are 1/2 and 1/2.

Now, Sum of zeroes = 1/2 + 1/2 = 1

= -(-4)/4 = -(Coefficient of s) / Coefficient of s²

Product of zeroes = 1/2 x 1/2 = 1/4

= 1/4 = Constant term / Coefficient of s².

(iii). 6x² – 3 – 7x

Solution:

= 6x² – 7x – 3

= 6x² – 9x + 2x – 3

= 3x(2x – 3) + 1(2x – 3)

= (3x + 1)(2x – 3)

The value of 6x² – 7x – 3 is zero, if 3x + 1 = 0 or 2x – 3 = 0.

=> x = -1/3 or x = 3/2.

Therefore, the zeroes of 6x² – 7x – 3 are -1/3 and 3/2.

Now, Sum of zeroes = -1/3 + 3/2

= (-2 + 9)/6 = 7/6 = -(-7)/6

= -(Coefficient of x) / Coefficient of x²

Product of zeroes = -1/3 x 3/2 = -1/2

= Constant term / Coefficient of x².

(iv). 4u² + 8u

Solution:

= 4u(u + 2)

The value of 4u² + 8u is zero if 4u = 0 or u + 2 = 0.

⇒ u = 0 or u = -2.

Therefore, the zeroes of 4u² + 8u are 0 and -2.

Now, Sum of zeroes = 0 + (-2) = -2

= -8/4 = -(Coefficient of u) / Coefficient of u²

Product of zeroes = 0 × (-2) = 0 = 0/4

= Constant term / Coefficient of u².

(v). t² – 15

Solution:

= t² – (√15)² = (t + √15)(t – √15)

The value of t² – 15 is zero if t + √15 = 0 or t – √15 = 0.

⇒ x = -√15 or x = √15.

Therefore, the zeroes of t² – 15 are -√15 and √15.

Now, Sum of zeroes = -√15 + √15 = 0 = -(0)/1

= -(Coefficient of t) / Coefficient of t²

Product of zeroes = (-√15) × √15 = -15 = -15/1

= Constant term / Coefficient of t².

(vi). 3x² – x – 4

Solution:

= 3x² – 4x + 3x – 4

= x(3x – 4) + 1(3x – 4)

= (3x – 4)(x + 1)

The value of 3x² – x – 4 is zero if 3x – 4 = 0 or x + 1 = 0.

⇒ x = 4/3 or x = -1.

Therefore, the zeroes of 3x² – x – 4 are 4/3 and -1.

Now, Sum of zeroes = 4/3 + (-1) = (4 – 3)/3 = 1/3

= -(-1)/3 = -(Coefficient of x) / Coefficient of x²

Product of zeroes = 4/3 × (-1) = -4/3 = -4/3

= Constant term / Coefficient of x².

2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

(i). 1/4, -1

Solution:

Let α and β are the zeroes of the polynomial ax² + bx + c, then we have

α + β = 1/4 = -b/a

αβ = -1 = -4/4 = c/a

On comparing,

a = 4, b = -1 and c = -4

Hence, the required quadratic polynomial is 4x² – x – 4.

(ii). √2, 1/3

Solution:

Let α and β are the zeroes of the polynomial ax² + bx + c, then we have

α + β = √2 = 3√2 / 3 = -b/a

αβ = 1/3 = c/a

On comparing,

a = 3, b = -3√2 and c = 1

Hence, the required quadratic polynomial is 3x² – 3√2x + 1.

(iii). 0, √5

Solution:

Let α and β are the zeroes of the polynomial ax² + bx + c, then we have

α + β = 0 = 0/1 = -b/a

αβ = √5 = √5/1 = c/a

On comparing,

a = 1, b = 0 and c = √5

Hence, the required quadratic polynomial is x² + 0.x + √5

Hence, the required quadratic polynomial is x² + √5.

(iv). 1, 1

Solution:

Let α and β are the zeroes of the polynomial ax² + bx + c, then we have

α + β = 1 = 1/1 = -b/a

αβ = 1 = 1/1 = c/a

On comparing,

a = 1, b = -1 and c = 1

Hence, the required quadratic polynomial is x² – x + 1.

(v). -1/4, 1/4

Solution:

Let α and β are the zeroes of the polynomial ax² + bx + c, then we have

α + β = -1/4 = -b/a

αβ = 1/4 = c/a

On comparing,

a = 4, b = 1 and c = 1

Hence, the required quadratic polynomial is 4x² + x + 1.

(vi). 4, 1

Solution:

Let α and β are the zeroes of the polynomial ax² + bx + c, then we have

α + β = 4 = 4/1 = -b/a

αβ = 1 = 1/1 = c/a

On comparing,

a = 1, b = -4 and c = 1

Hence, the required quadratic polynomial is x² – 4x + 1.

What are Polynomials?

According to Class 10 Maths Chapter 2 Polynomials chapter, A polynomial is an expression consisting of variables, coefficients, and exponents combined by addition, subtraction, or multiplication.

👉 Examples:

• 2x+32x + 3 → Linear polynomial
• x2+2x+5x^2 + 2x + 5 → Quadratic polynomial
• x3−7x+1x^3 – 7x + 1 → Cubic polynomial

👉 Key points:

• The degree of a polynomial is the highest power of the variable.
• Each single part like 2x,3×2,52x, 3x^2, 5 is called a term.
• Polynomials are classified as monomial (1 term), binomial (2 terms), and trinomial (3 terms).

Importance of Class 10 Maths Chapter 2 Polynomials

Why should you focus on this chapter?

• High Scoring: Questions come every year in CBSE exams.
• Foundation Builder: Helps in quadratic equations, calculus, and higher algebra.
• Real Applications: Used in physics, economics, and engineering.
• Easy to Practice: Formula-based, so with revision you can score full marks.

Key Concepts in Class 10 Maths Chapter 2 Polynomials

This Class 10 Maths Chapter 2 Polynomials chapter focuses on four major ideas:

1. Zeroes of a Polynomial

• The values of the variable for which the polynomial becomes zero.
• Example: For x2−9=0x^2 – 9 = 0, zeroes are x=3x = 3 and x=−3x = -3.

2. Geometrical Meaning of Zeroes

• Linear polynomial → 1 zero (line cuts x-axis once).
• Quadratic polynomial → 2, 1, or 0 zeroes depending on the parabola.
• Cubic polynomial → Up to 3 zeroes.

3. Relationship Between Zeroes and Coefficients

For quadratic polynomial ax2+bx+cax^2 + bx + c:

• Sum of zeroes = −b/a-b/a
• Product of zeroes = c/ac/a

For cubic polynomial ax3+bx2+cx+dax^3 + bx^2 + cx + d:

• Sum of zeroes = −b/a-b/a
• Sum of product of zeroes two at a time = c/ac/a
• Product of zeroes = −d/a-d/a

4. Division Algorithm

If p(x)p(x) is divided by g(x)g(x), then: p(x)=g(x)×q(x)+r(x)p(x) = g(x) \times q(x) + r(x)

where q(x)q(x) is quotient and r(x)r(x) is remainder

How to Prepare forClass 10 Maths Chapter 2 Polynomials

Here are some study tips:

• Step 1: Revise definitions and types of polynomials.
• Step 2: Practice graph questions from Exercise 2.1.
• Step 3: Memorize formulas of sum and product of zeroes.
• Step 4: Attempt Exercise 2.2 for coefficient relations.
• Step 5: Practice long division problems from Exercise 2.3.
• Step 6: Use 10th Maths NCERT Chapter 2 Solutions at Vidyayan Academy for clarity.

Common Mistakes to Avoid

Students often lose marks due to small errors:

• Confusing degree of polynomial (e.g., x2+0xx^2 + 0x still has degree 2).
• Forgetting the negative sign in formulas.
• Misreading graphs in Exercise 2.1.
• Skipping verification steps in Exercise 2.2.
• Leaving optional exercise, though it builds confidence.

Exam Preparation Tips

Preparing for Class 10 Maths Chapter 2 Polynomials becomes much easier when you follow a structured strategy. The first step is to revise all the solved examples from NCERT because they form the foundation of most exam questions. These examples cover different concepts like factorization, remainder theorem, and the division algorithm, which are crucial for exams.

Along with examples, students should practice Class 10 Maths Exercise 2.1 and Class 10 Maths Exercise 2.2 at least twice. Repetition helps in building confidence and ensures you don’t make silly mistakes during the final exam. While practicing the division algorithm questions, it is a good idea to time yourself. This helps in improving your speed and accuracy, which are both important for attempting the paper within the given time limit.

Another important tip is to write step-wise NCERT Solutions in exams. Remember that teachers award marks for each correct step, not just the final answer. So even if you make a small calculation mistake, you can still secure partial marks by presenting your solution neatly and logically.

Lastly, to polish your preparation, solve mock tests and sample questions using 10th Maths NCERT Chapter 2 Solutions from Vidyayan Academy. These solutions are explained in a student-friendly way and help you check whether your answers are correct. Practicing with these resources will give you the same feel as the actual exam.

Daily Study Plan for

A consistent daily routine is the key to mastering Class 10 Maths Chapter 2 Polynomials. On Day 1, begin with the basics—understand the meaning of polynomials, their degree, terms, and types. A strong foundation makes it easier to handle advanced exercises later.

On Day 2, move to NCERT Solution of Exercise 2.1, which focuses on representing polynomials through graphs. Spend enough time plotting and interpreting graphs, as visual understanding improves your clarity.

Day 3 should be dedicated to memorizing and applying important formulas, especially those used in NCERT Solution of Exercise 2.2. This exercise strengthens your understanding of factorization and identities, which are very common in exams.

By Day 6, revise the entire chapter with the help of NCERT Solutions from Vidyayan Academy. This revision ensures you recall every method and formula at the right time.

Finally, on Day 7, attempt a full chapter test without referring to notes. This self-assessment will show you your strong and weak areas before the actual exam.

🔹 Real-Life Applications of Class 10 Maths Chapter 2 Polynomials

The study of polynomials in Class 10 Maths Chapter 2 is not just limited to textbooks; it has many real-world applications. For example, in Physics, the equations of motion you learn are polynomial expressions. They help describe how objects move under certain conditions.

In Economics, functions like cost, revenue, and profit often follow polynomial models. Understanding polynomials allows economists to predict trends and make informed decisions.

In Engineering, polynomial equations are used while designing structures such as bridges, roads, and even aircraft. They help in calculating stress, load, and material strength, ensuring safety and efficiency.

In the modern world of Data Science, trend lines and regression analysis often use polynomial functions to model and predict real-life data. Whether it’s analyzing sales, weather patterns, or stock markets, polynomials play a crucial role.

Thus, by learning Class 10 Maths Chapter 2 Polynomials, students not only prepare for their exams but also build a foundation for advanced studies and careers in various fields.

🔹 Why Choose Vidyayan Academy?

At Vidyayan Academy, we provide:

• Step-by-step 10th Maths NCERT Chapter 2 Solutions.
• Free PDF notes for revision.
• Simple explanations in a student-friendly manner.
• Regular updates aligned with CBSE curriculum.

Our aim is to make Class 10 Maths Chapter 2 Polynomials simple, scoring, and interesting for every student.

🔹 Conclusion

To sum up, Class 10 Maths Chapter 2 Polynomials is one of the most scoring and conceptually rich topics in the NCERT syllabus. By understanding zeroes, their graphical meaning, coefficient relations, and the division algorithm, you can easily master this chapter.

Always practice Class 10 Maths Exercise 2.1 and 2.2 thoroughly, and use 10th Maths NCERT Chapter 2 Solutions from Vidyayan Academy to strengthen your preparation. With consistent practice and smart revision, you can secure full marks in this chapter

Why Choose Vidyayan Academy?

At Vidyayan Academy, we offer:

Detailed, step-by-step solutions for 10th Maths NCERT Chapter 2.

Free PDF notes for easy revision.

Clear explanations that are easy for students to understand.

Regular updates that align with the CBSE curriculum.

Our goal is to make Class 10 Maths Chapter 2 on Polynomials not only simple and scoring but also engaging for every student.

Conclusion

In summary, Class 10 Maths Chapter 2 on Polynomials is one of the most rewarding and conceptually rich topics in the NCERT syllabus. By grasping concepts like zeroes, their graphical interpretations, relationships between coefficients, and the division algorithm, you can master this chapter with ease.

Make sure to practice Class 10 Maths Exercises 2.1 and 2.2 thoroughly, and utilize the 10th Maths NCERT Chapter 2 Solutions from Vidyayan Academy to enhance your preparation. With consistent practice and smart revision strategies, you can aim for full marks in this chapter!