Chapter 2: Polynomials Class 10 Maths NCERT Solutions

Chapter 2: Polynomials — Exercise 2.1

Graphs aren’t just squiggly lines—they tell a story! This exercise is your secret weapon for scoring quick marks by learning how to “read” where a polynomial touches the ground.

Q1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

i) Graph (i)

The Logic: A “zero” of a polynomial is simply the point where the graph crosses or touches the x-axis. If the line stays away from the x-axis, it has no zeroes.
The Steps: Look at the horizontal line. It is parallel to the x-axis and never touches it.
The Result: The number of zeroes is 0.

ii) Graph (ii)

The Logic: Follow the curve with your finger. Every time you “hit” the horizontal x-axis, count it!
The Steps: The curve crosses the x-axis at exactly one point.
The Result: The number of zeroes is 1.

iii) Graph (iii)

Human Tip: Be careful! Don’t count the points where it hits the vertical y-axis. We only care about the horizontal x-axis for p(x).
The Steps: The curve crosses the x-axis at three distinct points.
The Result: The number of zeroes is 3.

iv) Graph (iv)

The Logic: This is a parabola (a U-shaped curve). Let’s see how many times it cuts the x-axis.
The Steps: The U-shape cuts the x-axis at two points.
The Result: The number of zeroes is 2.

v) Graph (v)

The Steps: Trace the “snake-like” curve. It intersects the x-axis at four different places.
The Result: The number of zeroes is 4.

vi) Graph (vi)

Pro-Tip: Sometimes the graph doesn’t “cross” the axis; it just “kisses” or touches it and goes back. That still counts as a zero!
The Steps: The curve intersects at one point and touches the x-axis at two other points.
The Result: The number of zeroes is 3.

Student Summary Table (Cheat Sheet)

Visual Observation	Meaning in Math	Number of Zeroes
No touch/intersection	No value of x makes p(x) = 0	0
Intersects once	One linear factor	1
Touches but doesn’t cross	Repeated zero at that point	Counts as a zero
Cuts x-axis ‘n’ times	Polynomial degree is at least ‘n’	n

Chapter 2: Polynomials — Exercise 2.2

This exercise is the “heart” of the chapter! Understanding how the zeros of a polynomial talk to its coefficients is a superpower that helps you solve quadratic equations in seconds.

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

i) x^2 – 2x – 8

The Logic: First, we find the “zeros” (the values of x that make the equation zero) using splitting the middle term. Then, we check if their Sum equals -b/a and their Product equals c/a.
The Steps:
1. Find Zeroes: x^2 – 4x + 2x – 8 = 0
2. x(x – 4) + 2(x – 4) = 0 => (x – 4)(x + 2) = 0
3. So, zeroes are 4 and -2. (Let’s call them alpha and beta).
4. Verify Sum: Alpha + Beta = 4 + (-2) = 2. Formula check: -b/a = -(-2)/1 = 2. Matches!
5. Verify Product: Alpha * Beta = 4 * (-2) = -8. Formula check: c/a = -8/1 = -8. Matches!
The Result: Zeroes are 4, -2. Relationship verified.

ii) 4s^2 – 4s + 1

Human Tip: Don’t let the ‘s’ scare you; it works exactly like ‘x’. This is a perfect square!
The Steps:
1. (2s – 1)^2 = 0
2. s = 1/2, 1/2.
3. Sum: 1/2 + 1/2 = 1. Formula: -(-4)/4 = 1.
4. Product: 1/2 * 1/2 = 1/4. Formula: 1/4.
The Result: Zeroes are 1/2, 1/2. Relationship verified.

iii) 6x^2 – 3 – 7x

Pro-Tip: Always rearrange the equation in the standard form (ax^2 + bx + c) first! So, write it as 6x^2 – 7x – 3.
The Steps:
1. Split -7x into -9x and 2x: 6x^2 – 9x + 2x – 3 = 0
2. 3x(2x – 3) + 1(2x – 3) = 0 => (3x + 1)(2x – 3) = 0
3. Zeroes: -1/3 and 3/2.
The Result: Zeroes are -1/3, 3/2. Relationship verified.

iv) 4u^2 + 8u

The Logic: No ‘c’ term? No problem! Just take the common factor out.
The Steps: 4u(u + 2) = 0. So, u = 0 or u = -2.
The Result: Zeroes are 0, -2. Relationship verified.

v) t^2 – 15

The Steps: t^2 = 15 => t = +root 15 or -root 15.
The Result: Zeroes are root 15, -root 15. Relationship verified.

Q2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively:

i) 1/4, -1

The Logic: Think of it like a recipe. If you have the Sum (S) and Product (P), the polynomial is always: x^2 – (Sum)x + (Product).
The Steps:
1. Polynomial = x^2 – (1/4)x + (-1)
2. Multiply by 4 to remove the fraction: 4x^2 – x – 4.
The Result: 4x^2 – x – 4

ii) root 2, 1/3

The Steps:
1. x^2 – (root 2)x + 1/3
2. Multiply by 3: 3x^2 – 3(root 2)x + 1.
The Result: 3x^2 – 3(root 2)x + 1

iii) 0, root 5

The Steps: x^2 – (0)x + root 5.
The Result: x^2 + root 5

Summary Table

Goal	Formula to Use	Common Mistake
Sum of Zeroes	-b / a	Forgetting the negative sign in -b!
Product of Zeroes	c / a	Using the wrong sign for ‘c’.
Build Polynomial	x^2 – (Sum)x + Product	Putting a plus sign before the Sum.
Standard Form	ax^2 + bx + c	Not rearranging terms before starting.

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Chapter 2: Polynomials Class 10 Maths NCERT Solutions

Contents

Class 10 Maths NCERT Solutions - Chapter wise

Chapter 1: Real Numbers