Assertion (A): The sum of the zeroes of the polynomial 3x^2 - 5x - 2 is 5/3. Reason (R): For a quadratic polynomial ax^2 + bx + c, the sum of zeroes is -b/a.

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

Class 10 Maths Chapter 2: Polynomials — Important Questions & Sample Paper

Q: How do I find the zeros of a quadratic polynomial?

Set the polynomial equal to zero and solve by factorisation (splitting the middle term) or use the quadratic formula: x = [-b ± √(b²-4ac)]/(2a). For instance, for 3x² – x – 4 = 0, factorisation yields (3x-4)(x+1)=0, so zeros are 4/3 and -1.

Q: What is the relationship between zeros and coefficients of a cubic polynomial?

For cubic ax³+bx²+cx+d with zeros α, β, γ: sum α+β+γ = -b/a, sum of products taken two at a time αβ+βγ+γα = c/a, and product αβγ = -d/a. This allows you to solve for unknown coefficients or construct a polynomial when zeros are known.

Q: How does the division algorithm assist in finding all zeros of a polynomial?

If some zeros (e.g., √3 and -√3) are given, then (x-√3)(x+√3)=x²-3 is a factor. Divide the polynomial by this factor to get a lower-degree quotient, then find its zeros. Together with the given zeros, you have all zeros of the original polynomial.

Q: What is a typical mistake while verifying the zero-coefficient relationship?

A common error is forgetting the negative sign in the sum formula: sum of zeros = -b/a, not b/a. Also, students might miscalculate the product or confuse the signs when substituting zeros into the relationship.

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Yes — this page has 44+ original Class 10 Mathematics Chapter 2 (“Polynomials”) important questions with answers (Multiple Choice (MCQ), Assertion–Reason, Short Answer, Short Answer, Long Answer, Case Study). Practise them free, or generate a full CBSE board-pattern sample paper (80 marks) and export it to PDF or Word — in English & Hindi, for 2026-27.

Class 10 CBSE Chapter 2, Polynomials, is a foundational topic in algebra that expands upon earlier knowledge of expressions. This chapter formally defines a polynomial's degree and zeros, covering linear, quadratic, and cubic forms. Students learn to find zeros by factorisation (splitting the middle term) or the quadratic formula for quadratics. A crucial skill is understanding and applying the relationship between zeros and coefficients: for quadratic ax^2+bx+c, sum of zeros = -b/a and product = c/a; for cubic ax^3+bx^2+cx+d, sum = -b/a, sum of pairwise products = c/a, and product = -d/a. The division algorithm for polynomials is another cornerstone: if p(x) and g(x) are polynomials with g(x) non-zero, there exist quotient q(x) and remainder r(x) such that p(x) = g(x)q(x) + r(x), with degree of r(x) < degree of g(x). This helps find all zeros when some are given. Typical exam questions ask students to determine zeros, verify zero-coefficient relationships, solve for unknown coefficients under given conditions (e.g., one zero is -1, or α–β=3), construct a polynomial from specified zeros, and use the division algorithm to factorise and find all zeros of higher-degree polynomials. Mastering these concepts is essential for board exams and advanced algebra.

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Mathematics — Polynomials

Class 10Time: 3 hrsMax Marks: 80

SECTION A

1.
If α and β are the zeroes of the quadratic polynomial 2x^2 - 5x - 3, then the value of α^2β + αβ^2 is:
(a) -15/4(b) 15/4(c) -15/2(d) 15/2
1
2.
If the polynomial p(x) = x^3 + ax^2 + bx + c has (x+1) as a factor, leaves remainder 4 when divided by (x-2), and p(1) = 0, then the value of a+b+c is:
(a) -2(b) -1(c) 0(d) 1
1
3.
If the zeroes of the quadratic polynomial x^2 + (a+1)x + b are 2 and -3, then the values of a and b are respectively:
(a) a=0, b=-6(b) a=2, b=-6(c) a=0, b=6(d) a=-2, b=6
1

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Marks distribution & blueprint

In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question type	Marks each	In our bank
Multiple Choice (MCQ)	1 mark	13
Assertion–Reason	1 mark	6
Short Answer	2 marks	8
Short Answer	3 marks	6
Long Answer	5 marks	5
Case Study	4 marks	6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

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Q1. If α and β are the zeroes of the quadratic polynomial 2x^2 - 5x - 3, then the value of α^2β + αβ^2 is:
1 mark
Multiple Choice (MCQ)
(A) -15/4(B) 15/4(C) -15/2(D) 15/2
▸ Answer▾ Answer
-15/4
Q2. If the polynomial p(x) = x^3 + ax^2 + bx + c has (x+1) as a factor, leaves remainder 4 when divided by (x-2), and p(1) = 0, then the value of a+b+c is:
1 mark
Multiple Choice (MCQ)
(A) -2(B) -1(C) 0(D) 1
▸ Answer▾ Answer
-1
Q3. If the zeroes of the quadratic polynomial x^2 + (a+1)x + b are 2 and -3, then the values of a and b are respectively:
1 mark
Multiple Choice (MCQ)
(A) a=0, b=-6(B) a=2, b=-6(C) a=0, b=6(D) a=-2, b=6
▸ Answer▾ Answer
a=0, b=-6
Q4. For what value of k will the polynomial x^2 - 7x + k have zeroes 3 and 4?
1 mark
Multiple Choice (MCQ)
(A) 7(B) 10(C) 12(D) 14
▸ Answer▾ Answer
12
Q5. Assertion (A): The sum of the zeroes of the polynomial 3x^2 - 5x - 2 is 5/3. Reason (R): For a quadratic polynomial ax^2 + bx + c, the sum of zeroes is -b/a.
1 mark
Assertion–Reason
(A) Both A and R are true and R is the correct explanation of A.(B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
▸ Answer▾ Answer
Both A and R are true and R is the correct explanation of A.
Q6. Find the sum and product of the zeros of the polynomial $5x^2 - 3x + 2$.
2 marks
Short Answer
▸ Answer▾ Answer
Sum = 3/5, Product = 2/5
Q7. The zeros of the cubic polynomial $x^3 - 12x^2 + 39x - 28$ are in arithmetic progression. Find the zeros.
2 marks
Short Answer
▸ Answer▾ Answer
1, 4, 7
Q8. If $(x+1)$ and $(x-2)$ are factors of $x^3 + ax^2 + bx - 6$, find $a$ and $b$.
3 marks
Short Answer
▸ Answer▾ Answer
$a = 2$, $b = -5$
Q9. Find the quadratic polynomial whose zeroes are $3 + 2\sqrt{2}$ and $3 - 2\sqrt{2}$.
3 marks
Short Answer
▸ Answer▾ Answer
$x^2 - 6x + 1$
Q10. On dividing the polynomial 2x^3 – 5x^2 + 7x – 1 by a polynomial g(x), the quotient and remainder are (2x – 3) and (2x + 2) respectively. Find g(x).
5 marks
Long Answer
▸ Answer▾ Answer
g(x) = x^2 – x + 1
Q11. If the zeroes of the cubic polynomial x^3 – 6x^2 + 11x – 6 are a – b, a, a + b, find the values of a and b.
5 marks
Long Answer
▸ Answer▾ Answer
a = 2, b = 1 or b = –1
Q12. While solving a cubic equation, Meena noticed that the polynomial $p(x) = x^3 - 6x^2 + 11x - 6$ has zeros that are in arithmetic progression.
4 marks
Case Study
1. (i) If the zeros are taken as $a-d, a, a+d$, find the value of $a$.2 marks
2. (ii) Hence, determine all three zeros.2 marks
▸ Answer▾ Answer
$a=2$; zeros: 1, 2, 3.

Frequently asked questions

How do I find the zeros of a quadratic polynomial?

Set the polynomial equal to zero and solve by factorisation (splitting the middle term) or use the quadratic formula: x = [-b ± √(b²-4ac)]/(2a). For instance, for 3x² – x – 4 = 0, factorisation yields (3x-4)(x+1)=0, so zeros are 4/3 and -1.

What is the relationship between zeros and coefficients of a cubic polynomial?