Class 10 Maths Chapter 2 – Polynomials | NCERT Solutions, Key Concepts & Board Questions (2025–26)

 Class 10 CBSE Maths – Chapter 2: Polynomials

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🔹 Key Concepts and Formulas

✅ What is a Polynomial?

A polynomial in one variable x is an expression of the form:
$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$
Where:

• $a_0, a_1, \ldots, a_n$ are real numbers

• n is a non-negative integer

• Each term is called a "monomial"

✅ Types of Polynomials Based on Degree:

• Zero Polynomial: 0

• Constant Polynomial: Degree = 0

• Linear Polynomial: Degree = 1 (e.g., $2x + 3$)

• Quadratic Polynomial: Degree = 2 (e.g., $x^2 + 2x + 1$)

• Cubic Polynomial: Degree = 3 (e.g., $x^3 - 6x^2 + 11x - 6$)

✅ Degree of a Polynomial:

The highest power of the variable in the polynomial.

✅ Zeroes/Roots of a Polynomial:

If $p(x)$ is a polynomial, and $p(a) = 0$, then $a$ is a zero or root of the polynomial.

✅ Geometrical Meaning:

The zero of a polynomial $p(x)$ is the x-coordinate where the graph intersects the x-axis.

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📘 NCERT Solved Examples

Example 1: Find p(0), p(1), and p(2) for $p(x) = x^2 - 3x + 2$

• $p(0) = 0^2 - 3 \times 0 + 2 = 2$

• $p(1) = 1^2 - 3 \times 1 + 2 = 1 - 3 + 2 = 0$

• $p(2) = 4 - 6 + 2 = 0$
  ✅ So, 1 and 2 are zeroes of the polynomial.

Example 2: Find the zero of $p(x) = ax + b$.

Let $p(x) = ax + b$. For zero:
$p(x) = 0 \Rightarrow ax + b = 0 \Rightarrow x = -\frac{b}{a}$

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🔹 📄 Exercise 2.1 – Solutions

Q1. Find the zero of the polynomial in each case:

(i) $p(x) = x + 5$ ⇒ zero = $-5$
(ii) $p(x) = x - 7$ ⇒ zero = $7$
(iii) $p(x) = 5x - 3$ ⇒ zero = $\frac{3}{5}$
(iv) $p(x) = 2x + 1$ ⇒ zero = $-\frac{1}{2}$
(v) $p(x) = \sqrt{2}x + 1$ ⇒ zero = $-\frac{1}{\sqrt{2}}$
(vi) $p(x) = x - \frac{1}{5}$ ⇒ zero = $\frac{1}{5}$

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🔹 📄 Exercise 2.2 – Solutions

Q1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients:

(i) $x^2 - 2x - 8$

• Factor: $(x - 4)(x + 2)$

• Zeroes: 4 and -2

• Sum = 2, Product = -8

• Verify: Sum = $-\frac{b}{a} = -(-2)/1 = 2$, Product = $c/a = -8/1 = -8$
  ✅ Verified

(ii) $x^2 + 4x + 4$

• Factor: $(x + 2)^2$

• Zeroes: -2 and -2

• Sum = -4, Product = 4
  ✅ Verified

(iii) $x^2 + x - 6$

• Factor: $(x + 3)(x - 2)$

• Zeroes: -3 and 2
  ✅ Verified

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🔹 📄 Exercise 2.3 – Solutions

Q1. Find a quadratic polynomial whose zeroes are:

(i) 4 and 5

• Required polynomial: $x^2 - (4+5)x + (4 \times 5) = x^2 - 9x + 20$

(ii) -2 and 1

• $x^2 + x - 2$

(iii) $\frac{1}{4}$ and $-\frac{1}{4}$

• $x^2 - (\frac{1}{4} - \frac{1}{4})x + (\frac{1}{4} \times -\frac{1}{4}) = x^2 - \frac{1}{16}$

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🔹 📄 Exercise 2.4 – Solutions

Q1. Divide the polynomial $p(x) = x^3 - 3x^2 + 5x - 3$ by $x - 1$

Use long division method:

• Divide $x^3$ by $x$: result = $x^2$

• Multiply $x^2(x - 1) = x^3 - x^2$

• Subtract: $(x^3 - 3x^2) - (x^3 - x^2) = -2x^2$

• Bring down next term: $-2x^2 + 5x$

• Divide $-2x^2$ by $x$: result = $-2x$

• Multiply: $-2x(x - 1) = -2x^2 + 2x$

• Subtract: $( -2x^2 + 5x ) - ( -2x^2 + 2x ) = 3x$

• Bring down: $3x - 3$

• Divide: $3x / x = 3$

• Multiply: $3(x - 1) = 3x - 3$

• Subtract: Remainder = 0
  ✅ Quotient = $x^2 - 2x + 3$

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🔹 🗌 Summary and Revision Notes

• A polynomial is an algebraic expression with variable powers as whole numbers.

• Degree = highest exponent

• Zeroes of a polynomial are the values of x for which the polynomial is zero.

• For a quadratic polynomial $ax^2 + bx + c$:

  • Sum of zeroes = $-\frac{b}{a}$

  • Product of zeroes = $\frac{c}{a}$

• If zeroes are known: form = $x^2 - (sum)x + product$

• Division Algorithm: $p(x) = d(x) \cdot q(x) + r(x)$, where degree of r(x) < degree of d(x)

📘 CBSE Past Year Questions – Chapter: Polynomials

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✅ 1-Mark Questions

1. (2020) Find the zero of the polynomial:
   $p(x) = 5x + 3$
   📌 Answer: $x = -\frac{3}{5}$

2. (2019) Write a polynomial whose zero is $\frac{3}{2}$.
   📌 Answer: $p(x) = 2x - 3$

3. (2017) If one zero of the quadratic polynomial $3x^2 - 2x + k = 0$ is $\frac{1}{3}$, find the value of $k$.
   📌 Answer: Use the relation:
   Let $\alpha = \frac{1}{3}, \beta = ?$,
   $\alpha + \beta = \frac{2}{3} \Rightarrow \beta = \frac{2}{3} - \frac{1}{3} = \frac{1}{3}$
   So both roots are $\frac{1}{3}$, apply in:
   $\alpha\beta = \frac{k}{3} \Rightarrow \frac{1}{9} = \frac{k}{3} \Rightarrow k = \frac{1}{3}$

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✅ 2-Mark Questions

1. (2020) If the zeroes of a quadratic polynomial $x^2 + px + 12$ are 3 and -4, find the value of p.
   📌 Answer:
   Sum of zeroes = 3 + (–4) = –1
   $-\frac{b}{a} = -\frac{p}{1} \Rightarrow -1 = -p \Rightarrow p = 1$

2. (2018) If one zero of the polynomial $x^2 + 2x + k$ is 3, find the value of k.
   📌 Answer:
   Let the other zero be $\beta$.
   Sum = 3 + $\beta = -2 \Rightarrow \beta = -5$
   Product = $3 \times -5 = -15 = k$

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✅ 3-Mark Questions

1. (2023) Find the zeroes of the quadratic polynomial $x^2 - 7x + 12$ and verify the relationship between zeroes and coefficients.
   📌 Answer:
   Factor: $(x - 3)(x - 4)$ → zeroes: 3, 4
   Sum = 7, Product = 12
   Verify: $-\frac{b}{a} = 7, \frac{c}{a} = 12$ ✅

2. (2020) Find a quadratic polynomial whose sum and product of zeroes are √2 and 1/3 respectively.
   📌 Answer:
   Polynomial: $x^2 - (\text{sum})x + \text{product} = x^2 - \sqrt{2}x + \frac{1}{3}$

3. (2016) Find the zeroes of the polynomial $x^2 + 5x + 6$ and verify the relationship between the zeroes and coefficients.
   📌 Answer:
   $x^2 + 5x + 6 = (x + 2)(x + 3)$ → zeroes: –2, –3
   Sum = –5, Product = 6
   Verify: $-b/a = -5/1 = -5, c/a = 6$ ✅

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✅ 4-Mark Questions

1. (2019) Divide the polynomial $p(x) = x^3 - 6x^2 + 11x - 6$ by $g(x) = x - 1$ and verify the division algorithm:
   📌 Answer:

   • Quotient = $x^2 - 5x + 6$, Remainder = 0

   • Verify: $p(x) = g(x) × q(x) + r(x)$

2. (2021 Sample) Find a quadratic polynomial whose zeroes are reciprocal of each other and the sum is 10.
   📌 Answer:
   Let zeroes = $\alpha, \frac{1}{\alpha}$,
   Sum = $\alpha + \frac{1}{\alpha} = 10$
   Multiply both sides by $\alpha$: $\alpha^2 + 1 = 10\alpha \Rightarrow \alpha^2 - 10\alpha + 1 = 0$
   Polynomial = $x^2 - 10x + 1$

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✅ 5-Mark Questions

1. (2020 – Internal Assessment)
   A cubic polynomial is divided by a linear polynomial. Use long division to find quotient and remainder and verify the division algorithm.
   $p(x) = x^3 - 3x^2 + 5x - 3$, divisor = $x - 1$
   📌 Answer:
   Long division gives:
   Quotient = $x^2 - 2x + 3$, Remainder = 0
   Verify: $p(x) = (x - 1)(x^2 - 2x + 3) + 0$

2. (CBSE 2017 – Sample)
   If the sum and product of zeroes of a quadratic polynomial are –1 and 1 respectively, construct the polynomial and verify zeroes.
   📌 Answer:
   Polynomial: $x^2 + x + 1$
   Zeroes via quadratic formula:
   $x = \frac{-1 \pm \sqrt{1^2 - 4(1)(1)}}{2} = \frac{-1 \pm \sqrt{-3}}{2}$ → imaginary roots

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🔁 Topics Frequently Asked in Exams:

• Finding zeroes of polynomials

• Verifying relationships between zeroes and coefficients

• Forming quadratic polynomials from given zeroes

• Division of polynomials and application of Division Algorithm

• Conceptual 2-mark questions involving identities

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