CBSE Class 10 Mathematics • Chapter 2
Polynomials Class 10 Worksheet with Solutions and MCQs
Practise finding zeroes, understanding polynomial graphs and applying the relationship between zeroes and coefficients. Use this Class 10 Polynomials worksheet alongside the explanations, solved examples and extra questions below to strengthen your algebra skills.
A Polynomials Class 10 worksheet with solutions helps you connect formulas with the steps needed to solve a question. Instead of memorising only the sum and product of zeroes, learn how to identify coefficients, factorise an expression and verify your answer. These skills are useful when working through NCERT exercises, classroom assignments and Class 10 Maths revision questions.
If you are looking for a worksheet on Polynomials Class 10 with answers, begin with the worksheet available on this page. The additional practice below includes quadratic polynomials, alpha beta questions, finding the value of k, MCQs and a case-based example. Try each question independently before reading its solution.
Polynomials Class 10 Introduction: Definition and Examples
A polynomial in one variable x is an expression made from terms such as axn, where the exponent n is a non-negative integer and the coefficient a is a real number. The highest power of x with a non-zero coefficient is the degree of a non-zero polynomial.
Which Expressions Are Polynomials?
Examples: 6x, x2 − 5x + 6, 3x3 + 2x − 1 and 9 are polynomials.
Non-examples: 1/x and √x are not polynomials in x. Their variable exponents are −1 and 1/2 respectively, which are not non-negative integers.
The expression √2x + 3 is a polynomial because √2 is a real coefficient and the power of x is 1.
| Type of Polynomial | Degree | Example |
|---|---|---|
| Non-zero constant polynomial | 0 | 9 |
| Linear polynomial | 1 | 6x − 12 |
| Quadratic polynomial | 2 | x2 − 5x + 6 |
| Cubic polynomial | 3 | 2x3 − x + 4 |
Important Topics for Class 10 Polynomials Practice
Zeroes of a Polynomial
A number k is a zero of p(x) when p(k) = 0. For example, 2 is a zero of p(x) = x − 2 because p(2) = 0.
Geometrical Meaning of Zeroes
The real zeroes are the x-coordinates of the points where the graph y = p(x) meets the x-axis. Count distinct meeting points, including points where the graph touches the axis.
Relationship Between Zeroes and Coefficients
For a quadratic polynomial ax2 + bx + c, use the coefficients to find the sum and product of its zeroes.
Forming a Quadratic Polynomial
When the sum S and product P of two zeroes are given, one suitable quadratic polynomial is x2 − Sx + P.
Polynomials Class 10 Formulas for Alpha Beta Questions
Let α and β be the zeroes of p(x) = ax2 + bx + c, where a ≠ 0. This formula sheet brings together the relationships commonly used in Class 10 Polynomials extra questions.
| Relationship | Formula |
|---|---|
| Sum of zeroes | α + β = −b/a |
| Product of zeroes | αβ = c/a |
| Sum of squares of zeroes | α2 + β2 = (α + β)2 − 2αβ |
| Square of the difference | (α − β)2 = (α + β)2 − 4αβ |
| Sum of reciprocals, when αβ ≠ 0 | 1/α + 1/β = (α + β)/(αβ) |
| A polynomial with zeroes α and β | x2 − (α + β)x + αβ |
Remember the Signs
In x2 − 5x + 6, the coefficient b is −5. Therefore, the sum of zeroes is −(−5)/1 = 5. Write a, b and c with their signs before substituting them into a formula.
Class 10 Polynomials Important Questions with Solutions
The following original examples cover useful question types for revision. They are additional practice questions, rather than a collection labelled as previous year board questions.
1. Find the Zeroes and Verify Their Relationship
Question: Find the zeroes of 2x2 − 7x + 3 and verify the relationship between zeroes and coefficients.
Solution:
2x2 − 7x + 3
= 2x2 − 6x − x + 3
= 2x(x − 3) − 1(x − 3)
= (2x − 1)(x − 3).
Hence, the zeroes are 1/2 and 3.
Sum: 1/2 + 3 = 7/2 = −(−7)/2 = −b/a.
Product: (1/2) × 3 = 3/2 = c/a.
Both relationships are verified.
2. Form a Quadratic Polynomial from Its Zeroes
Question: Find a quadratic polynomial whose zeroes have sum −3 and product −10.
Solution:
A suitable polynomial is x2 − Sx + P.
Substituting S = −3 and P = −10 gives
x2 + 3x − 10.
Check: x2 + 3x − 10 = (x + 5)(x − 2). Its zeroes are −5 and 2, with sum −3 and product −10.
3. Find the Value of k
Question: If 2 is a zero of x2 + kx − 6, find k.
Solution: Since 2 is a zero, p(2) = 0.
4 + 2k − 6 = 0
2k = 2, so k = 1.
The polynomial becomes x2 + x − 6 = (x − 2)(x + 3). Its other zero is −3.
4. Solve an Alpha Beta Identity Question
Question: If α and β are the zeroes of 3x2 − 5x + 1, find α2 + β2 without finding the individual zeroes.
Solution:
α + β = 5/3 and αβ = 1/3.
α2 + β2
= (α + β)2 − 2αβ
= 25/9 − 2/3
= 19/9.
Polynomials Class 10 MCQ with Answers
Attempt these objective questions before opening the answers. Each explanation shows the concept behind the correct option.
MCQ 1: Degree of a Polynomial
What is the degree of 5x3 − 2x + 7?
A. 1 B. 2 C. 3 D. 7
Show Answer
Answer: C. 3. The highest power of x with a non-zero coefficient is 3.
MCQ 2: Sum of Zeroes
What is the sum of the zeroes of 4x2 + 8x − 3?
A. 2 B. −2 C. −3/4 D. 3/4
Show Answer
Answer: B. −2. The sum is −b/a = −8/4 = −2.
MCQ 3: Zeroes from a Graph
A polynomial graph meets the x-axis at x = −2 and x = 3 only. How many distinct real zeroes does it have?
A. 0 B. 1 C. 2 D. 3
Show Answer
Answer: C. 2. The two distinct x-intercepts represent the zeroes −2 and 3.
MCQ 4: Forming a Polynomial
Which quadratic polynomial has zeroes 4 and −1?
A. x2 + 3x − 4
B. x2 − 3x − 4
C. x2 − 5x + 4
D. x2 + 5x + 4
Show Answer
Answer: B. The sum is 3 and the product is −4. Therefore, x2 − 3x − 4 is a suitable polynomial.
Polynomials Class 10 Case Study Questions with Solutions
Case Study: A Rectangular Display Board
A rectangular display board has length (x + 3) metres and width (x + 2) metres, where x > 0. Its area is represented by a polynomial A(x).
Question 1: Write the area polynomial.
A(x) = (x + 3)(x + 2)
= x2 + 5x + 6.
Question 2: Find the algebraic zeroes of A(x).
From the factors, the zeroes are −3 and −2.
Question 3: Verify the sum and product of the zeroes.
Sum = −3 + (−2) = −5 = −b/a.
Product = (−3)(−2) = 6 = c/a.
Question 4: Find the area when x = 2.
A(2) = 4 + 10 + 6 = 20 square metres.
Interpretation: The negative zeroes are valid algebraically, but they lie outside the given physical condition x > 0. They do not represent usable dimensions for this display board.
Class 10 Polynomials Extra Questions for Self-Practice
- Find the zeroes of x2 − 9x + 20 and verify their sum and product.
- Form a quadratic polynomial whose zeroes are −2 and 5.
- If −1 is a zero of 2x2 + kx − 3, find k.
- If α and β are the zeroes of x2 − 6x + 7, find α2 + β2.
- Find the sum of the reciprocals of the zeroes of 2x2 − 7x + 3.
Check the Answers
- Zeroes: 4 and 5; sum: 9; product: 20.
- One suitable polynomial: x2 − 3x − 10.
- k = −1.
- α2 + β2 = 36 − 14 = 22.
- Sum of reciprocals = (7/2)/(3/2) = 7/3.
How to Use This Polynomials Class 10 Practice Worksheet
Start with the definition, degree and meaning of zeroes. Next, practise factorisation and verification questions before attempting alpha beta identities and case-based questions. This sequence helps you understand why a formula works and when to use it.
For Polynomials Class 10 Exercise 2.1, focus on the geometrical meaning of zeroes and reading x-intercepts. For Exercise 2.2, practise finding zeroes, verifying their relationship with coefficients and forming quadratic polynomials. Follow the exercise numbering in your own NCERT edition.
You can also use these extra questions as a short Class 10 Polynomials test. Solve them without your formula sheet, check the answers and revisit any incorrect steps. When practising Polynomials Class 10 previous year questions or NCERT Exemplar questions separately, apply the same method: identify the concept, show the calculation and verify the result.
Common Mistakes to Avoid
- Using b/a instead of −b/a for the sum of zeroes.
- Forgetting to divide by a when the leading coefficient is not 1.
- Counting y-intercepts instead of x-intercepts in graph questions.
- Assuming that every quadratic polynomial has two distinct real zeroes.
- Using reciprocal formulas when one of the zeroes is 0.
- Writing only the final zeroes when the question also asks for verification.
FAQs on Polynomials Class 10 Worksheet and Solutions
What is a polynomial? Explain with a Class 10 example.
A polynomial in x contains terms whose powers of x are non-negative integers. For example, 2x2 − 7x + 3 is a quadratic polynomial because its highest power of x is 2. The expression 1/x + 3 is not a polynomial in x.
Is 9 a polynomial? What is its degree?
Yes. The number 9 is a non-zero constant polynomial of degree 0 because it can be written as 9x0. The zero polynomial is different: its degree is not defined.
How do I find the zeroes of a quadratic polynomial?
Set the polynomial equal to zero and factorise it when possible. For example, x2 − 5x + 6 = (x − 2)(x − 3). Setting each factor equal to zero gives the zeroes 2 and 3. Check them by substituting each value into the original polynomial.
How do I verify the relationship between zeroes and coefficients?
For ax2 + bx + c, first find the zeroes α and β. Calculate α + β and compare it with −b/a. Then calculate αβ and compare it with c/a. Show both comparisons in your solution.
What are the important Polynomials Class 10 alpha beta formulas?
The main formulas are α + β = −b/a and αβ = c/a. For extra questions, use α2 + β2 = (α + β)2 − 2αβ. If both zeroes are non-zero, their reciprocal sum is (α + β)/(αβ).
How can I form a polynomial when the sum and product of zeroes are given?
If the sum is S and the product is P, one suitable quadratic polynomial is x2 − Sx + P. For example, sum 5 and product 6 give x2 − 5x + 6. Any non-zero constant multiple of this polynomial has the same zeroes.
How do I solve Polynomials Class 10 find the value of k questions?
If a zero is given, substitute it into the polynomial and set the result equal to zero. For example, if 2 is a zero of x2 + kx − 6, then 4 + 2k − 6 = 0, giving k = 1. If the sum or product of zeroes is given instead, use the coefficient relationships.
How many real zeroes can a quadratic polynomial have?
A quadratic polynomial can have zero, one or two distinct real zeroes. The graph of x2 + 1 has no x-intercepts; x2 touches the x-axis at one point; and x2 − 1 meets it at two points.
Does touching the x-axis count as a zero?
Yes. A zero occurs whenever p(x) = 0, whether the graph crosses or touches the x-axis. For example, (x − 2)2 touches the x-axis at x = 2. It has one distinct real zero, repeated twice.
Can I solve alpha beta questions without finding the actual zeroes?
Yes. Many expressions can be calculated from the sum and product. For example, if α + β = 6 and αβ = 7, then α2 + β2 = 36 − 14 = 22. Finding α and β individually is unnecessary for this question.
Which Class 10 Polynomials questions should I practise for revision?
Practise finding zeroes, verifying coefficient relationships, forming quadratic polynomials, reading graphs, finding unknown coefficients and using alpha beta identities. Include MCQs and case-based questions so that you can apply the concepts in different formats.
How should I use a Polynomials Class 10 worksheet PDF with solutions?
Attempt the questions first and use the solutions afterwards to check your method. Mark errors involving signs, factorisation or formulas. Revise the relevant concept, then solve the incorrect question again without looking at the answer.
Where can I find Polynomials Class 10 notes and a mind map?
Visit the Polynomials Class 10 Notes and Mind Map page on WitKnowLearn. Use it to review the concepts before attempting this practice worksheet.
What is a polynomial called in Hindi?
Polynomial को हिंदी में बहुपद कहते हैं। उदाहरण के लिए, x2 − 5x + 6 एक द्विघात बहुपद है। किसी बहुपद का शून्यक वह मान होता है जिसे चर के स्थान पर रखने पर बहुपद का मान शून्य हो जाता है।
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