Polynomials Class 10 Previous Year Questions (Chapter 2)

Polynomials in CBSE board papers

CBSE asked 175 different Polynomials questions in the Class 10 Mathematics board papers of 2023, 2024, 2025 and 2026 (main and compartment exams, Standard and Basic courses). 7 of these questions have a worked solution on this page. Most of them test Section 2.3, Relationship between Zeroes and Coefficients of a Polynomial (148 of 175).

Where Polynomials questions came from, 2023–2026

A question printed in several question-paper sets counts once in each table. A question asked in two exams counts in both rows of the first table.

Questions by exam

Year	Exam	Questions asked	Marks	Question types
2026	Main	27	9 × 1 mark · 8 × 2 marks · 9 × 3 marks · 1 × 4 marks	8 MCQs, 1 Assertion–Reason, 1 case study and 17 written answers
2025	Main	41	14 × 1 mark · 6 × 2 marks · 21 × 3 marks	14 MCQs and 27 written answers
2025	Compartment	11	4 × 1 mark · 4 × 2 marks · 3 × 3 marks	3 MCQs and 8 written answers
2024	Main	47	29 × 1 mark · 6 × 2 marks · 10 × 3 marks · 2 × 4 marks	26 MCQs, 3 Assertion–Reason, 2 case studies and 16 written answers
2024	Compartment	9	3 × 1 mark · 6 × 3 marks	2 MCQs, 1 Assertion–Reason and 6 written answers
2023	Main	42	33 × 1 mark · 3 × 2 marks · 5 × 3 marks · 1 × 4 marks	29 MCQs, 3 Assertion–Reason, 1 case study and 9 written answers
2023	Compartment	9	5 × 1 mark · 3 × 3 marks · 1 × 4 marks	5 MCQs, 1 case study and 3 written answers

Questions by NCERT section and marks

NCERT section	1 mark	2 marks	3 marks	4 marks	Total
2.2 Geometrical Meaning of the Zeroes of a Polynomial	21	0	0	5	26
2.3 Relationship between Zeroes and Coefficients of a Polynomial	74	27	47	0	148
Mixed topics	1	0	0	0	1
All questions	96	27	47	5	175

Most-tested sections

2.3 Relationship between Zeroes and Coefficients of a Polynomial: 148 questions
2.2 Geometrical Meaning of the Zeroes of a Polynomial: 26 questions

From 2023 to 2026, CBSE took most Polynomials questions from Section 2.3 (148 of 175). These counts describe past papers; they are not predictions.

Topics in Polynomials

Each NCERT section of the chapter, in book order: what CBSE asked from it and where to practise it.

2.2 Geometrical Meaning of the Zeroes of a Polynomial

A real number k is a zero of p(x) if p(k) = 0. For a linear polynomial ax + b, a ≠ 0, the graph meets the x-axis at exactly one point; its x-coordinate is the zero. For a quadratic ax² + bx + c, a ≠ 0, zeroes are the x-coordinates where its parabola meets the x-axis. A polynomial of degree n has at most n zeroes.

How to answer questions on this section

Count the points where the graph y = p(x) intersects the x-axis.
For a linear polynomial ax + b, locate the point (-b/a, 0) on the x-axis.
For a quadratic, check whether the parabola cuts, touches, or misses the x-axis.
For a cubic, count x-axis intersections; there can be at most three zeroes.
For degree n, state the graph can meet the x-axis at at most n points.

Mistakes to avoid in this section

Counting where the graph meets the y-axis instead of the x-axis.
Saying a graph touching the x-axis at one point gives two zeroes; it gives one.
Treating a turning point as a zero when the curve does not meet the x-axis there.

CBSE asked 26 questions from this section in the 2023–2026 papers (21 of 1 mark; 5 of 4 marks): 15 MCQs, 5 Assertion–Reason, 5 case studies and 1 written answer. 17 were printed in Standard papers and 9 in Basic papers.

Marks	Questions asked	Years	Asked in Standard	Asked in Basic
1 mark	21	2023, 2024, 2025, 2026	13	8
4 marks	5	2023, 2024, 2026	4	1

2.3 Relationship between Zeroes and Coefficients of a Polynomial

For linear ax + b, a ≠ 0, the zero is −b/a. Let ax² + bx + c, a ≠ 0, have zeroes α, β. Then sum = −b/a and product = c/a. Let ax³ + bx² + cx + d, a ≠ 0, have zeroes α, β, γ. Then sum = −b/a, sum of products two at a time = c/a, product = −d/a.

How to answer questions on this section

For quadratic ax² + bx + c, factorise by splitting the middle term.
Use factors of ac whose sum is b to write the split.
Equate each factor to zero; the solutions are zeroes.
Verify zeroes satisfy sum = −b/a and product = c/a.
For cubics, check sum, products two at a time, and product = −d/a.

Mistakes to avoid in this section

While splitting the middle term, taking product as c instead of ac.
Writing sum of zeroes as b/a, forgetting the negative sign.
When forming a polynomial, writing x² + Sx + P instead of x² − Sx + P.

CBSE asked 148 questions from this section in the 2023–2026 papers (74 of 1 mark; 27 of 2 marks; 47 of 3 marks): 71 MCQs, 2 Assertion–Reason and 75 written answers. 75 were printed in Standard papers and 73 in Basic papers.

Marks	Questions asked	Years	Asked in Standard	Asked in Basic
1 mark	74	2023, 2024, 2025, 2026	43	31
2 marks	27	2023, 2024, 2025, 2026	14	13
3 marks	47	2023, 2024, 2025, 2026	18	29

Solved on this page: question 1, question 2, question 3, question 4, question 5, question 6, question 7.

1 more question uses several sections at once; the tables above count it under Mixed topics.

Key ideas used on this page

Relationship between Zeroes and Coefficients of a Polynomial
Used in questions 1 2 3 5 6 7
Relationship between zeroes and coefficients of a quadratic polynomial
Used in question 4

5 solved questions last asked in 2026

Q1. Find a quadratic polynomial whose sum and product of zeroes are 0 and -9, respectively. Also, find the zeroes of the polynomial so obtained.

Mark practised

Marks: 3
Level: Basic
Asked in: 2 paper sets
Years: 2025, 2026
Topic: 2.3 Relationship between Zeroes and Coefficients of a Polynomial
Source: (CBSE 2026 Mathematics Basic, Set 430/1/1, Q27, 3 marks)
Status: Answer checked step by step

Step-by-step solution

Step 1: Form the polynomial
1 mark
Let α and β be the zeroes. Given α + β = 0 and αβ = −9. A quadratic polynomial with these zeroes is x² − (α + β)x + αβ = x² − 0x − 9 = x² − 9.
Why? For a quadratic polynomial ax² + bx + c, the sum of its zeroes is −b/a and the product is c/a. So a polynomial with zeroes α and β can be written as x² − (α + β)x + αβ.
Step 2: Factorise the polynomial
1 mark
x² − 9 = (x − 3)(x + 3).
Why? The identity a² − b² = (a − b)(a + b) applies because 9 = 3².
Step 3: Write the zeroes
1 mark
So x = 3 or x = −3. The quadratic polynomial is x² − 9, and its zeroes are 3 and −3.
Why? A number k is a zero of p(x) if p(k) = 0. A product of factors is zero when at least one factor is zero, so x − 3 = 0 or x + 3 = 0.

Marks: 1 + 1 + 1 = 3 · Split as in the CBSE marking scheme, 2025 · Basic 430/1/1 Q27 (the marks split is taken from the 2025 Basic paper, where this question also appeared)

Answer:

x² − 9; 3, −3

Common mistakes to avoid

Writing x² + 9 instead of x² − 9; the product is negative, so the constant term must be negative.
Giving only 3 as a zero and missing −3; x² − 9 has two factors, (x − 3) and (x + 3), so both x = 3 and x = −3 make the polynomial zero.
Stopping after writing x² − 9 and not finding the zeroes, although the question asks for both the polynomial and its zeroes.

Why this works

The relationship between zeroes and coefficients of a quadratic polynomial says that if α and β are the zeroes of ax² + bx + c, then α + β = −b/a and αβ = c/a. So a quadratic polynomial whose zeroes have sum S and product P can be written as x² − Sx + P. Here the sum is 0 and the product is −9, so the polynomial is x² − 9. To find its zeroes, use the difference of squares identity: x² − 9 = (x − 3)(x + 3). The zeroes are the values that make a factor zero, so they are 3 and −3. Quick check: 3 + (−3) = 0 and 3 × (−3) = −9, which match the given sum and product. Multiplying the polynomial by any non-zero real number gives all quadratic polynomials that fit the conditions, as NCERT Example 4 shows.

Key idea to remember

Relationship between Zeroes and Coefficients of a Polynomial: see Key ideas used on this page.

Explanation

NCERT proves that if α and β are the zeroes of ax² + bx + c, a ≠ 0, then α + β = −b/a and αβ = c/a. Therefore a quadratic polynomial with these zeroes can be written as x² − (α + β)x + αβ. Substituting the sum 0 and the product −9 gives x² − 9. Factorising x² − 9 as (x − 3)(x + 3) gives the zeroes 3 and −3.

Important formulas

x² − (α + β)x + αβ · Quadratic polynomial whose zeroes are α and β.
α + β = −b/a, αβ = c/a · Relationship between zeroes and coefficients for ax² + bx + c, a ≠ 0.
a² − b² = (a − b)(a + b) · Identity used to factorise x² − 9.

Q2. Find the zeroes of the polynomial 25a² - 10a + 1 and verify the relationship between the zeroes and coefficients of the given polynomial.

Mark practised

Marks: 3
Level: Basic
Asked in: 2 paper sets
Years: 2025, 2026
Topic: 2.3 Relationship between Zeroes and Coefficients of a Polynomial
Source: (CBSE 2026 Mathematics Basic, Set 430/3/3, Q29, 3 marks)
Status: Answer checked step by step

Step-by-step solution

Step 1: Factorise the polynomial
1 mark
25a² - 10a + 1 = (5a - 1)², so 5a - 1 = 0, giving a = 1/5.
Why? A product is zero when one of its factors is zero; here both factors are identical, so the same zero repeats.
Step 2: Verify sum of zeroes
1 mark
Sum of zeroes = 1/5 + 1/5 = 2/5. Also, -b/a = -(-10)/25 = 2/5.
Why? For a quadratic polynomial ax² + bx + c, the sum of its zeroes equals -b/a.
Step 3: Verify product of zeroes
1 mark
Product of zeroes = 1/5 × 1/5 = 1/25. Also, c/a = 1/25. Hence the zeroes are 1/5 and 1/5.
Why? For a quadratic polynomial ax² + bx + c, the product of its zeroes equals c/a.

Marks: 1 + 1 + 1 = 3 · Split as in the CBSE marking scheme, 2025 · Basic 430/3/3 Q29 (the marks split is taken from the 2025 Basic paper, where this question also appeared)

Answer:

The zeroes are 1/5 and 1/5.

Common mistakes to avoid

Writing only one zero, 1/5, because the zeroes are equal; a quadratic polynomial can have two equal zeroes, and here they are 1/5 and 1/5.
Verifying the product as the constant term 1 instead of c/a = 1/25.
Using (5a + 1)², which makes the middle term +10a, while the given polynomial has -10a.

Why this works

The polynomial is a perfect square, so factorising it as (5a - 1)² is the quickest way to find the zeroes. A product is zero only when one of its factors is zero, so 5a - 1 = 0 gives a = 1/5. The factor appears twice, so the two zeroes are both 1/5. The relationship between zeroes and coefficients says that for 25a² - 10a + 1, whose coefficients are 25, -10 and 1, the sum of the zeroes is -b/a = 2/5 and the product is c/a = 1/25. These match 1/5 + 1/5 and 1/5 × 1/5. Splitting the middle term is another correct method.

Key idea to remember

Relationship between Zeroes and Coefficients of a Polynomial: see Key ideas used on this page.

Explanation

For a quadratic polynomial ax² + bx + c (a ≠ 0), if α and β are its zeroes, then α + β = -b/a and αβ = c/a. Here the two zeroes are both 1/5, so their sum is 2/5 and their product is 1/25, which matches the values -b/a and c/a.

Important formulas

α + β = -b/a · For a quadratic polynomial ax² + bx + c, a ≠ 0.
αβ = c/a · For a quadratic polynomial ax² + bx + c, a ≠ 0.
(a - b)² = a² - 2ab + b² · Used to factorise 25a² - 10a + 1 as (5a - 1)².

Q3. Find the quadratic polynomial the sum of whose zeroes is 1 and their product is -12. Hence find the zeroes of the polynomial.

Mark practised

Marks: 2
Level: Standard
Years: 2026
Topic: 2.3 Relationship between Zeroes and Coefficients of a Polynomial
Source: (CBSE 2026 Mathematics Standard, Set 30(B), Q22, 2 marks)
Status: Verified solution

Step-by-step solution

Step 1: Form the polynomial
1 mark (suggested)
Let p(x) = x² + bx + c, taking the coefficient of x² as 1. Given sum of zeroes is 1 and product of zeroes is −12. Sum = −b/a gives b = −1. Product = c/a gives c = −12. Hence p(x) = x² − x − 12.
Why? For a quadratic polynomial ax² + bx + c, the sum of its zeroes is −b/a and the product is c/a. Choosing the coefficient of x² as 1 lets us read b and c directly.
Step 2: Find the zeroes
1 mark (suggested)
x² − x − 12 = (x − 4)(x + 3), so x = 4 or x = −3. Hence the zeroes of the polynomial are 4 and −3.
Why? If the product of two factors is 0, then one of the factors must be 0.

Marks: 1 + 1 = 2 · Suggested split (CBSE’s usual pattern for 2-mark questions)

Answer:

x² − x − 12; 4, −3

✓ Answer matches the CBSE marking scheme, 2026 Main, Paper 30(B).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing x² + x − 12 because the sum is 1, forgetting that the coefficient of x is −(sum of zeroes), so b = −1.
Taking the zeroes as 4 and 3, since 4 × 3 = 12, but the product must be −12, so one zero must be negative.
Stopping after writing the polynomial and not factorising to give the zeroes, so the answer is incomplete.

Why this works

The question gives the sum and product of the zeroes. NCERT Example 4 shows the rule: for p(x) = ax² + bx + c, sum of zeroes = −b/a and product of zeroes = c/a. Taking the coefficient of x² as 1 gives b = −(sum) = −1 and c = product = −12, so p(x) = x² − x − 12. To find the zeroes, factorise: x² − x − 12 = (x − 4)(x + 3). A quick check: 4 + (−3) = 1 and 4 × (−3) = −12, which match the given sum and product. Also 4² − 4 − 12 = 0 and (−3)² + 3 − 12 = 0, so both values are zeroes.

Key idea to remember

Relationship between Zeroes and Coefficients of a Polynomial: see Key ideas used on this page.

Explanation

For a quadratic polynomial ax² + bx + c, if α and β are its zeroes, then α + β = −b/a and αβ = c/a. This solution uses these relations to build the polynomial, then factorises it to find the zeroes.

Important formulas

α + β = −b/a · Sum of zeroes of ax² + bx + c.
αβ = c/a · Product of zeroes of ax² + bx + c.
x² − (α + β)x + αβ · Quadratic polynomial with coefficient of x² as 1.

Q4. Find the zeroes of the quadratic polynomial \(x^2 + 7x + 10\) , and verify the relationship between the zeroes and its coefficients.

Mark practised

Marks: 2
Level: Standard
Years: 2026
Topic: 2.3 Relationship between Zeroes and Coefficients of a Polynomial
Source: (CBSE 2026 Mathematics Standard, Set 30/1/2, Q25, 2 marks)
Status: Verified solution

Step-by-step solution

Step 1: Factorise and find the zeroes
1 mark (suggested)
x² + 7x + 10 = (x + 2)(x + 5), so x + 2 = 0 or x + 5 = 0, giving x = -2 or x = -5.
Why? A product of two factors is zero only when at least one factor is zero, so each factor gives a zero of the polynomial.
Step 2: Verify the relationship
1 mark (suggested)
Sum of zeroes = -2 + (-5) = -7 = -7/1 = -(coefficient of x)/(coefficient of x²). Product of zeroes = (-2)(-5) = 10 = 10/1 = constant term/(coefficient of x²). Hence the zeroes are -2 and -5, and the relationship is verified.
Why? For ax² + bx + c, the sum of zeroes is -b/a and the product is c/a; matching these with the computed sum and product verifies the relationship.

Marks: 1 + 1 = 2 · Suggested split (CBSE’s usual pattern for 2-mark questions)

Answer:

The zeroes are -2 and -5.

✓ Answer matches the CBSE marking scheme, 2026 Main, Paper 30/1/2.

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing the zeroes as 2 and 5 because (x + 2) and (x + 5) are the factors, without solving x + 2 = 0 and x + 5 = 0.
Verifying the sum as 7 instead of -7, because the formula uses -b/a, not b/a.
Splitting 7x as 1x + 10x because 1 × 10 = 10, forgetting that 1 + 10 = 11, not 7.
Writing the product as -10 because one zero is negative, forgetting that (-2)(-5) = 10.

Why this works

To factorise x² + 7x + 10 by splitting the middle term, find two numbers whose sum is 7 and whose product is 10; 2 and 5 work, since 2 + 5 = 7 and 2 × 5 = 10. Hence x² + 7x + 10 = (x + 2)(x + 5). A product is zero only when one factor is zero, so x + 2 = 0 gives x = -2 and x + 5 = 0 gives x = -5. Quick check: (-2)² + 7(-2) + 10 = 4 - 14 + 10 = 0, and (-5)² + 7(-5) + 10 = 25 - 35 + 10 = 0. For verification, put the polynomial in the form ax² + bx + c; then the sum of the zeroes should be -b/a and the product should be c/a. Here the coefficient of x² is 1, so -b/a = -7/1 and c/a = 10/1. The zeroes give sum = -2 + (-5) = -7 and product = (-2)(-5) = 10, which match -7/1 and 10/1.

Key idea to remember

Relationship between zeroes and coefficients of a quadratic polynomial: see Key ideas used on this page.

Explanation

For a quadratic polynomial ax² + bx + c, if alpha and beta are its zeroes, then alpha + beta = -b/a and alpha beta = c/a. This solution factorises x² + 7x + 10 to get the zeroes -2 and -5, then checks that their sum -2 + (-5) = -7 equals -7/1 = -b/a and their product (-2)(-5) = 10 equals 10/1 = c/a.

Important formulas

alpha + beta = -b/a · Sum of the zeroes of ax² + bx + c.
alpha beta = c/a · Product of the zeroes of ax² + bx + c.

Q5. Find a quadratic polynomial whose zeroes are \((5 - 2\sqrt{3})\) and \((5 + 2\sqrt{3})\).

Mark practised

Marks: 2
Level: Standard
Years: 2026
Topic: 2.3 Relationship between Zeroes and Coefficients of a Polynomial
Source: (CBSE 2026 Mathematics Standard, Set 30/1/3, Q24, 2 marks)
Status: Verified solution

Step-by-step solution

Step 1: Find the sum and product of the zeroes
1 mark (suggested)
Let the zeroes be α and β. Then α + β = (5 − 2√3) + (5 + 2√3) = 10 and αβ = (5 − 2√3)(5 + 2√3) = 25 − 12 = 13.
Why? For a quadratic polynomial, the sum and product of its zeroes are related to its coefficients; here we compute them from the given zeroes, using (a − b)(a + b) = a² − b² for the product.
Step 2: Write the quadratic polynomial
1 mark (suggested)
The required quadratic polynomial is x² − (α + β)x + αβ = x² − 10x + 13.
Why? A quadratic polynomial with leading coefficient 1 is x² − (sum of zeroes)x + (product of zeroes); substituting the computed sum and product gives the required polynomial.

Marks: 1 + 1 = 2 · Suggested split (CBSE’s usual pattern for 2-mark questions)

Answer:

x² − 10x + 13

✓ Answer matches the CBSE marking scheme, 2026 Main, Paper 30/1/3.

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing x² + 10x + 13 because the sign of the sum is taken as positive; the formula has −(α + β), so the coefficient of x is −10.
Taking the product as the sum of the squares, 25 + 12, instead of subtracting them; the correct product is 13.
Using the zeroes as 5 ± 2√3 but writing the constant term as −13; the product of the two zeroes is positive 13.

Why this works

Given the zeroes, use the relationship between zeroes and coefficients. Sum = (5 − 2√3) + (5 + 2√3) = 10. Product = (5 − 2√3)(5 + 2√3) = 25 − 12 = 13. For a quadratic polynomial with leading coefficient 1, x² − (sum)x + product = x² − 10x + 13. Any k(x² − 10x + 13), k real, also has these zeroes; the polynomial with leading coefficient 1 is one such polynomial. Another correct method is to multiply (x − (5 − 2√3))(x − (5 + 2√3)) directly; it gives the same polynomial.

Key idea to remember

Relationship between Zeroes and Coefficients of a Polynomial: see Key ideas used on this page.

Explanation

If α and β are zeroes of ax² + bx + c, then α + β = −b/a and αβ = c/a. When the coefficient of x² is 1, the polynomial is x² − (α + β)x + αβ. This solution finds the sum and product of the given zeroes and substitutes them into this form.

Important formulas

α + β = −b/a · Sum of the zeroes of ax² + bx + c.
αβ = c/a · Product of the zeroes of ax² + bx + c.
x² − (α + β)x + αβ · Quadratic polynomial with leading coefficient 1 and zeroes α, β.
(a − b)(a + b) = a² − b² · Used to multiply the two given zeroes.

1 solved question last asked in 2024

Q6. If one zero of the polynomial \(x^2 - 8x + k\) exceeds the other by 2, then find the zeroes and the value of \(k\).

Mark practised

Marks: 3
Level: Basic
Years: 2024
Topic: 2.3 Relationship between Zeroes and Coefficients of a Polynomial
Source: (CBSE 2024 Mathematics Basic, Set 430/3/3, Q31, 3 marks)
Status: Verified solution

Step-by-step solution

Step 1: Form the relations
1 mark (suggested)
Let the zeroes be α and β, with α = β + 2. For x² − 8x + k, α + β = 8 and αβ = k.
Why? For a quadratic ax² + bx + c, a and b are the coefficients (the numbers multiplying x² and x) and c is the constant term. The sum of the zeroes is −b/a and the product is c/a, so the coefficients give two equations for the unknown zeroes.
Step 2: Solve for the zeroes
1 mark (suggested)
Put α = β + 2 in α + β = 8: (β + 2) + β = 8, so β = 3 and α = 5.
Why? Substituting one relation into the other gives a linear equation in one unknown; solving it fixes both zeroes.
Step 3: Find k and state the result
1 mark (suggested)
k = αβ = 5 × 3 = 15. Hence the zeroes are 5 and 3, and k = 15.
Why? The product of the zeroes of a quadratic is c/a; here the coefficient of x² is one, so this product equals k.

Marks: 1 + 1 + 1 = 3 · Suggested split (CBSE’s usual pattern for 3-mark questions)

Answer:

The zeroes are 5 and 3, and k = 15.

✓ Answer matches the CBSE marking scheme, 2024 Main, Paper 430/3/3.

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing the condition as α = β − 2 when α is the larger zero; this reverses the two zeroes.
Using the sum as −8 instead of 8; the formula gives sum = −(−8)/1 = 8.
Finding the zeroes 5 and 3 but then taking k as 8, the sum, instead of 15, the product.
Not checking that p(5) and p(3) are zero after choosing k.

Why this works

Here the unknown zeroes are linked in two ways: the polynomial tells us their sum and product, and the question tells us their difference. For x² − 8x + k, comparing with ax² + bx + c gives sum = −b/a = −(−8)/1 = 8 and product = c/a = k. Let the larger zero be α and the smaller be β. Then α = β + 2. Put this in α + β = 8: (β + 2) + β = 8, so 2β = 6 and β = 3. Then α = 3 + 2 = 5. Finally k = αβ = 5 × 3 = 15. Quick check: 5 + 3 = 8, 5 − 3 = 2, 5 × 3 = 15, and putting the zeroes into x² − 8x + 15 gives 25 − 40 + 15 = 0 and 9 − 24 + 15 = 0. Another correct method also earns full marks.

Key idea to remember

Relationship between Zeroes and Coefficients of a Polynomial: see Key ideas used on this page.

Explanation

For a quadratic polynomial ax² + bx + c with zeroes α and β, the sum of the zeroes is −b/a and the product is c/a. In this question, the given condition gives the difference of the zeroes, and the sum is read from the coefficient of x. Solving the sum and difference together gives the zeroes, and the product gives k.

Important formulas

α + β = −b/a · For a quadratic polynomial ax² + bx + c, the sum of the zeroes.
αβ = c/a · For a quadratic polynomial ax² + bx + c, the product of the zeroes.

1 solved question last asked in 2023

Q7. Find the zeroes of the quadratic polynomial \(x^2 + 6x + 8\) and verify the relationship between the zeroes and the coefficients.

Mark practised

Marks: 3
Level: Basic
Asked in: 2 paper sets
Years: 2023
Topic: 2.3 Relationship between Zeroes and Coefficients of a Polynomial
Source: (CBSE 2023 Mathematics Basic, Set 430/4/1, Q26, 3 marks)
Status: Verified solution

Step-by-step solution

Step 1: Factorise the polynomial
1 mark (suggested)
x² + 6x + 8 = (x + 2)(x + 4)
Why? Splitting the middle term works because two numbers with the required sum and product let the quadratic be grouped into two identical factors.
Step 2: Find the zeroes
1 mark (suggested)
The value is zero when x + 2 = 0 or x + 4 = 0, so x = −2 or x = −4.
Why? A zero of p(x) is a number k for which p(k) = 0; since a product is zero only when one factor is zero, each factor gives one zero.
Step 3: Verify the relationship
1 mark (suggested)
Sum of zeroes = (−2) + (−4) = −6 = −6/1 = −b/a. Product of zeroes = (−2) × (−4) = 8 = 8/1 = c/a. Hence the zeroes are −2 and −4, and the relationship is verified.
Why? For p(x) = ax² + bx + c with zeroes α and β, the relations α + β = −b/a and αβ = c/a hold; substituting the found zeroes into these relations confirms them.

Marks: 1 + 1 + 1 = 3 · Suggested split (CBSE’s usual pattern for 3-mark questions)

Answer:

−2 and −4

✓ Answer matches the CBSE marking scheme, 2023 Main, Paper 430/4/1.

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing the zeroes as 2 and 4, because the sign is not changed when solving x + 2 = 0 and x + 4 = 0.
Writing the sum of zeroes as 6, the coefficient of x, instead of −6.
Writing only the constant term 8 for the product, without showing that it equals c/a.

Why this works

The polynomial is x² + 6x + 8. Find two numbers whose sum is 6 and product is 8; they are 2 and 4, so x² + 6x + 8 = (x + 2)(x + 4). A product is zero only when one factor is zero, so the zeroes are −2 and −4. Compare with ax² + bx + c: the coefficient relations give sum of zeroes = (−2) + (−4) = −6 = −b/a and product of zeroes = (−2)(−4) = 8 = c/a. So the relationship holds. Quick check: (−2)² + 6(−2) + 8 = 4 − 12 + 8 = 0, and (−4)² + 6(−4) + 8 = 16 − 24 + 8 = 0.

Key idea to remember

Relationship between Zeroes and Coefficients of a Polynomial: see Key ideas used on this page.

Explanation

For a quadratic polynomial ax² + bx + c, if α and β are its zeroes, then p(x) = k(x − α)(x − β). Comparing coefficients gives α + β = −b/a and αβ = c/a. This solution finds the zeroes by factorisation and then checks these two relations.

Important formulas

α + β = −b/a · For p(x) = ax² + bx + c, the sum of the zeroes is the negative of the coefficient of x divided by the coefficient of x².
αβ = c/a · The product of the zeroes is the constant term divided by the coefficient of x².
x² + (p + q)x + pq = (x + p)(x + q) · Used to factorise the given quadratic by splitting the middle term.

Most repeated Polynomials questions

The questions CBSE printed in the most question-paper sets, 2023–2026. Each one is solved where the line points.

Printed in 2 question-paper sets (2025 and 2026) · 3 marks · Basic · question 1 on this page
Printed in 2 question-paper sets (2025 and 2026) · 3 marks · Basic · question 2 on this page
Printed in 2 question-paper sets (2023) · 3 marks · Basic · question 7 on this page

Official papers behind this chapter

Polynomials was asked in 120 of the 120 question-paper sets of 2023–2026. The table names the official CBSE files for each exam in which it was asked.

The board publishes the question papers on cbse.gov.in and the marking schemes on cbse.gov.in, where you can see how marks are given.

Year	Exam	Course	Sets with a Polynomials question	Question paper	Marking scheme
2026	Main	Standard (041)	10 of 10 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper.	Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 22.8 MB, Class 10 Maths Standard (041)) to see how marks are given.
2026	Main	Basic (241)	10 of 10 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper.	Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 29.1 MB, Class 10 Maths Basic (241)) to see how marks are given.
2025	Main	Standard (041)	17 of 17 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 54.5 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Main	Basic (241)	12 of 12 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 49.9 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Compartment	Standard (041)	3 of 3 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 3.4 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 8.1 MB, Class 10 Maths Standard (041)) to see how marks are given.
2025	Compartment	Basic (241)	3 of 3 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 2.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 18.7 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Main	Standard (041)	14 of 14 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 35.3 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2024	Main	Basic (241)	14 of 14 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 43.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.8 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Compartment	Standard (041)	3 of 3 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2024 compartment exam (one ZIP file of about 3.2 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2024 compartment exam (one ZIP file of about 7.1 MB, Class 10 Maths Standard (041)) to see how marks are given.
2024	Compartment	Basic (241)	4 of 4 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2024 compartment exam (one ZIP file of about 2.2 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 compartment exam (one ZIP file of about 5.6 MB, Class 10 Maths Basic (241)) to see how marks are given.
2023	Main	Standard (041)	12 of 12 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 44 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 17.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2023	Main	Basic (241)	12 of 12 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 49.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 26 MB, Class 10 Maths Basic (241)) to see how marks are given.
2023	Compartment	Standard (041)	3 of 3 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2023 compartment exam (one ZIP file of about 7.6 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 compartment exam (one ZIP file of about 3.3 MB, Class 10 Maths Standard (041)) to see how marks are given.
2023	Compartment	Basic (241)	3 of 3 question-paper sets had a Polynomials question	Open CBSE's official question paper for the 2023 compartment exam (one ZIP file of about 7.2 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2023 compartment exam (one ZIP file of about 7.5 MB, Class 10 Maths Basic (241)) to see how marks are given.

Continue studying

Other chapters: Real Numbers previous year questions.

To revise the lessons behind these questions, read NCERT Chapter 2 (official PDF), the chapter exactly as NCERT publishes it on ncert.nic.in.

Sources and data verification

The counts on this page come from 120 CBSE Class 10 Mathematics question-paper sets: 62 Standard (041) and 58 Basic (241), from the main exams of 2023, 2024, 2025 and 2026 and the compartment exams of 2023, 2024 and 2025. A question printed in several sets counts once; these counts describe past papers and are not predictions.

CBSE settles the question papers and the marking schemes: each question is reproduced as printed in its CBSE paper, with its year, paper code and question number, and each answer shown here matches the official marking scheme or answer key. NCERT settles the textbook: chapter and section names follow NCERT Mathematics Class X (reprint 2026-27).

Every solution is original. The mark beside each step follows the official CBSE marking scheme where the scheme shows it; a mark labelled “(suggested)” follows the board’s usual pattern for that type of question.

The official question-paper and marking-scheme links on this page were checked and open on 9 October 2026.

Reference: CBSE Class 10 Mathematics board question papers via cbse.gov.in; all solutions are original. learncbse.net is an independent website and is not affiliated with CBSE or NCERT.

Polynomials in CBSE board papers

Where Polynomials questions came from, 2023–2026

Questions by exam

Questions by NCERT section and marks

Most-tested sections

Topics in Polynomials

2.2 Geometrical Meaning of the Zeroes of a Polynomial

How to answer questions on this section

Mistakes to avoid in this section

2.3 Relationship between Zeroes and Coefficients of a Polynomial

How to answer questions on this section

Mistakes to avoid in this section

Key ideas used on this page

5 solved questions last asked in 2026

Q1. Find a quadratic polynomial whose sum and product of zeroes are 0 and -9, respectively. Also, find the zeroes of the polynomial so obtained.

Step-by-step solution

Step 1: Form the polynomial

Step 2: Factorise the polynomial

Step 3: Write the zeroes

Answer:

Common mistakes to avoid

Q2. Find the zeroes of the polynomial 25a² - 10a + 1 and verify the relationship between the zeroes and coefficients of the given polynomial.

Step-by-step solution

Step 1: Factorise the polynomial

Step 2: Verify sum of zeroes

Step 3: Verify product of zeroes

Answer:

Common mistakes to avoid

Q3. Find the quadratic polynomial the sum of whose zeroes is 1 and their product is -12. Hence find the zeroes of the polynomial.

Step-by-step solution

Step 1: Form the polynomial

Step 2: Find the zeroes

Answer:

Common mistakes to avoid

Q4. Find the zeroes of the quadratic polynomial \(x^2 + 7x + 10\) , and verify the relationship between the zeroes and its coefficients.

Step-by-step solution

Step 1: Factorise and find the zeroes

Step 2: Verify the relationship

Answer:

Common mistakes to avoid

Q5. Find a quadratic polynomial whose zeroes are \((5 - 2\sqrt{3})\) and \((5 + 2\sqrt{3})\).

Step-by-step solution

Step 1: Find the sum and product of the zeroes

Step 2: Write the quadratic polynomial

Answer:

Common mistakes to avoid

1 solved question last asked in 2024

Q6. If one zero of the polynomial \(x^2 - 8x + k\) exceeds the other by 2, then find the zeroes and the value of \(k\).

Step-by-step solution

Step 1: Form the relations

Step 2: Solve for the zeroes

Step 3: Find k and state the result

Answer:

Common mistakes to avoid

1 solved question last asked in 2023

Q7. Find the zeroes of the quadratic polynomial \(x^2 + 6x + 8\) and verify the relationship between the zeroes and the coefficients.

Step-by-step solution

Step 1: Factorise the polynomial

Step 2: Find the zeroes

Step 3: Verify the relationship

Answer:

Common mistakes to avoid

Most repeated Polynomials questions

Official papers behind this chapter

Continue studying

Sources and data verification

More from this section