CBSE Class 10 Mathematics: NCERT Solutions for Chapter 2 Polynomials (2025-2026 Guide)

For students preparing for CBSE Class 10 Mathematics in the 2025-2026 academic session, mastering NCERT Solutions Class 10 Maths Chapter 2 Polynomials is essential. Polynomials form the algebraic backbone for higher secondary mathematics, laying the groundwork for quadratic equations, coordinate geometry, and calculus. This chapter carries significant weight in the board exam 10 paper, featuring direct graphical questions, coefficient-zero verifications, and algebraic evaluations.

In this guide, we provide a complete breakdown of core definitions, theorems, geometrical interpretations, and step-by-step solutions to high-yield CBSE board exam questions aligned with the latest rationalized NCERT syllabus.

Key Concepts and Formulas: Chapter 2 Polynomials

A polynomial in one variable x is an algebraic expression of the form:

p(x) = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀

where a_n, a_n-1, …, a₀ are real constants with a_n ≠ 0, and n is a non-negative integer (whole number).

1. Degree of a Polynomial

The degree of a polynomial p(x) is the highest exponent of the variable x in p(x).

• Linear Polynomial: A polynomial of degree 1. Standard form: p(x) = ax + b, where a ≠ 0. It has at most 1 real zero given by x = -b/a.
• Quadratic Polynomial: A polynomial of degree 2. Standard form: p(x) = ax² + bx + c, where a ≠ 0. It has at most 2 real zeroes.
• Cubic Polynomial: A polynomial of degree 3. Standard form: p(x) = ax³ + bx² + cx + d, where a ≠ 0. It has at most 3 real zeroes.

2. Geometrical Meaning of the Zeroes of a Polynomial

A real number k is called a zero of a polynomial p(x) if p(k) = 0.

Geometrically, the zeroes of y = p(x) are precisely the x-coordinates of the points where the graph intersects or touches the x-axis.

• Linear graphs: The graph of y = ax + b is a straight line intersecting the x-axis at exactly one point: (-b/a, 0).
• Quadratic graphs: The graph of a quadratic polynomial y = ax² + bx + c is a U-shaped curve called a parabola.
  • If a > 0, the parabola opens upwards (∪).
  • If a < 0, the parabola opens downwards (∩).
• Number of Zeroes: A polynomial p(x) of degree n intersects the x-axis at a maximum of n distinct points. Hence, it can have at most n real zeroes.

3. Relationship Between Zeroes and Coefficients

Let α and β be the zeroes of the quadratic polynomial p(x) = ax² + bx + c (where a ≠ 0). The algebraic relationships connecting the roots to the coefficients are:

1. Sum of Zeroes: α + β = -b / a = -(Coefficient of x) / (Coefficient of x²)
2. Product of Zeroes: αβ = c / a = (Constant term) / (Coefficient of x²)

4. Forming a Quadratic Polynomial from Given Zeroes

If the sum of zeroes S = α + β and the product of zeroes P = αβ are known, the family of quadratic polynomials is given by:

p(x) = k [x² - (α + β)x + αβ] = k [x² - Sx + P]

where k is any non-zero real constant.

5. Important Algebraic Identities for Zeroes

CBSE board exam questions frequently ask students to evaluate symmetric expressions involving α and β. Keep these standard expansions handy:

• α² + β² = (α + β)² - 2αβ = S² - 2P
• 1/α + 1/β = (α + β) / (αβ) = S / P
• (α - β)² = (α + β)² - 4αβ = S² - 4P
• α³ + β³ = (α + β)³ - 3αβ(α + β) = S³ - 3PS

6. Real-World Applications of Polynomials

Polynomials model numerous real-world physical and architectural phenomena:

• Projectile Motion: The path of a kicked football, a basketball arching toward a hoop, or water emerging from a fountain follows a downward parabolic curve modeled by a quadratic polynomial h(t) = -gt² + v₀t + h₀.
• Civil Engineering: Parabolic arches in bridges and roller coasters distribute load efficiently across structural pillars.
• Economics and Profit Maximization: Revenue, cost, and profit functions are routinely expressed as quadratic or cubic polynomials to identify break-even points and optimal output levels.

Important CBSE Questions with Step-by-Step Solutions

Below are essential questions from official CBSE question banks and NCERT exercises designed to test core competencies for CBSE Mathematics.

Question 1 (Graphical Understanding — 1 Mark)

Question: The graph of y = p(x) is given in a coordinate plane. If the curve intersects the x-axis at points (-3, 0), (1, 0), and (4, 0), find the number of zeroes of p(x) and state the zeroes.

Solution:

1. The zeroes of a polynomial y = p(x) are the x-coordinates of the points where the graph intersects the x-axis.
2. The curve intersects the x-axis at 3 distinct points: x = -3, x = 1, and x = 4.

Answer: The polynomial has 3 zeroes, and the zeroes are -3, 1, and 4.

---

Question 2 (Zeroes & Verification — 2/3 Marks)

Question: Find the zeroes of the quadratic polynomial p(x) = 6x² - 3 - 7x and verify the relationship between the zeroes and the coefficients.

Solution:

Step 1: Write in standard form

p(x) = 6x² - 7x - 3

Comparing with ax² + bx + c, we get: a = 6, b = -7, c = -3.

Step 2: Factorize by splitting the middle term

We need two numbers whose product is a × c = 6 × (-3) = -18 and sum is b = -7. The numbers are -9 and +2.

6x² - 9x + 2x - 3 = 0
3x(2x - 3) + 1(2x - 3) = 0
(2x - 3)(3x + 1) = 0

Setting each factor to zero:

2x - 3 = 0 ⇒ x = 3/2
3x + 1 = 0 ⇒ x = -1/3

Therefore, the zeroes are α = 3/2 and β = -1/3.

Step 3: Verify the relationship

• Sum of Zeroes:
  LHS: α + β = 3/2 + (-1/3) = (9 - 2)/6 = 7/6
  RHS: -b / a = -(-7) / 6 = 7/6
  LHS = RHS (Verified)
• Product of Zeroes:
  LHS: αβ = (3/2) × (-1/3) = -3/6 = -1/2
  RHS: c / a = -3 / 6 = -1/2
  LHS = RHS (Verified)

---

Question 3 (Forming a Quadratic Polynomial — 2 Marks)

Question: Find a quadratic polynomial whose sum and product of zeroes are 1/4 and -1 respectively.

Solution:

Let the zeroes of the quadratic polynomial be α and β.

Given: Sum of zeroes (S) = α + β = 1/4
Product of zeroes (P) = αβ = -1

The general equation of a quadratic polynomial is:

p(x) = k [x² - (Sum of zeroes)x + (Product of zeroes)]
p(x) = k [x² - (1/4)x + (-1)]
p(x) = k [x² - x/4 - 1]

Taking k = 4 to clear the denominator:

p(x) = 4(x² - x/4 - 1) = 4x² - x - 4

Answer: The required quadratic polynomial is 4x² - x - 4.

---

Question 4 (Evaluating Symmetric Expressions — 3 Marks)

Question: If α and β are the zeroes of the polynomial f(x) = 2x² - 5x + 7, evaluate: (i) α² + β² and (ii) 1/α + 1/β.

Solution:

For the polynomial f(x) = 2x² - 5x + 7, we have a = 2, b = -5, c = 7.

Using the coefficient relations:

α + β = -b / a = -(-5) / 2 = 5/2
αβ = c / a = 7/2

Part (i): Evaluate α² + β²

We know: α² + β² = (α + β)² - 2αβ

α² + β² = (5/2)² - 2(7/2) = 25/4 - 7 = (25 - 28) / 4 = -3/4

Part (ii): Evaluate 1/α + 1/β

1/α + 1/β = (α + β) / (αβ) = (5/2) / (7/2) = 5/7

Answer: (i) -3/4, (ii) 5/7.

---

Question 5 (Finding Unknown Parameter k — 3 Marks)

Question: If the sum of the squares of zeroes of the quadratic polynomial p(x) = x² - 8x + k is 40, find the value of k.

Solution:

Let α and β be the zeroes of p(x) = x² - 8x + k.

Here, a = 1, b = -8, c = k.

Sum of zeroes: α + β = -(-8)/1 = 8
Product of zeroes: αβ = k/1 = k

We are given that α² + β² = 40.

Using the identity α² + β² = (α + β)² - 2αβ:

40 = (8)² - 2(k)
40 = 64 - 2k
2k = 64 - 40
2k = 24 ⇒ k = 12

Answer: The value of k is 12.

How to Prepare for This Topic in CBSE Class 10 Board Exam

Achieving a 100% score in Polynomials requires conceptual clarity and strict attention to presentation details during your board exam 10 revision:

1. Rearrange Terms into Standard Form First: Examiners frequently scramble terms (e.g., writing 6x² - 3 - 7x instead of 6x² - 7x - 3). Always reorder terms descending by degree before identifying a, b, and c to avoid incorrect sign assignments.
2. Beware of the Negative Sign in Sum of Roots: The formula is -(b/a). When b itself is negative (such as b = -5), the expression becomes -(-5)/a = +5/a. Common calculation errors occur here.
3. Always Include the Constant k in Formed Polynomials: When constructing a quadratic polynomial from given zeroes, write p(x) = k(x² - Sx + P) and specify the choice of k that clears any fractional denominators.
4. Master Middle-Term Factorization: Practice quadratic polynomials with square root coefficients (e.g., 4√3x² + 5x - 2√3) as they are standard 3-mark questions in the CBSE board exam.
5. Verify Your Working: After calculating the zeroes, quickly check that substituting them back into the original expression yields zero.

Where to Practice More

To secure full marks in CBSE Class 10 Mathematics, regular practice with authentic, board-pattern test papers is indispensable. Practice chapter-wise question sets, access previous years' solved board papers, and generate customized mock exams on the QPTool Question Bank. You can also explore targeted assessments through Theorify QPTool Class 10 Practice Tests and benchmark your preparation with the latest CBSE Class 10 Mathematics Sample Papers.