#### Class ten maths Chap 2 Polynomials NCERT Solution

## Chapter 2: Polynomials NCERT Solutions

**TEXTBOOK’S EXERCISE – 2.1
**

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

**Sol. **(i) From the graph, it is clear that it does not intersect the x-axis. Hence, it has no zero.

(ii) From the graph, it is clear that it intersects the x-axis at one point. Hence, it has one zero.

(iii) From the graph, it is clear that it intersects the x-axis at three points. Hence, it has three zeroes.

(iv) From the graph, it is clear that it intersects the x-axis at two points. Hence, it has two zeroes.

(v) From the graph, it is clear that it intersects the x-axis at four points. Hence, it has four zeroes.

(vi) From the graph, it is clear that it intersects the x-axis at three points. Hence, it has three zeroes.

**TEXTBOOK’S EXERCISE – 2.2 **

**1. **Find the zeroes of
the following quadratic polynomials and verify the relationship between the
zeroes and the coefficients.

(i) x^{2} – 2x – 8

(ii) 4s^{2} – 4s + 1

(iii) 6x^{2} – 3 – 7x

(iv) 4u^{2} + 8u

(v) t^{2} – 15

(vi) 3x^{2} – x – 4

**Sol. **(i) Quadratic polynomial is

X^{2} – 2x – 8 = x^{2}–4x + 2x – 8

=X( x–4) + 2(x–4)

=(x–4) (x–2)

The value of x^{2}–2x – 8 is
zero, if

(x – 4) = 0 or (x + 2) = 0

Then, x = 4 and x = – 2.

So, zeroes of x^{2} – 2x – 8 are –2 and 4

Thus, sum of zeroes = (–2) + (4) = 2

Product of zeroes = (–2) (4) = – 8

Hence, relationship between the zeroes and the coefficients is verified.

(ii) Quadratic polynomial is

4s^{2 }–
4s+ 1= 4s^{2}– 2s-2s+1

=2s(2s-1) – (2s-1)

= (2s-1) (2s-1)

This value of 4s^{2 }– 4s+ 1 is zero, if (2s-1) = 0 or (2s-1)=0

Product of zeroes

Hence, relationship between the zeroes and the coefficients is verified.

(iii) Quadratic polynomial is

6x^{2} – 3- 7x = 6x^{2} – 7x -3
= 6x^{2} – 9x +2x – 3

=3x (2x- 3) +1(2x-3)= (2x-3) (3x +1)

The value of 6x^{2}-3 -7x is zero, if

(2x-3) =0 or (3x+1)=0

Then x = 3/2 or x = −1/3

So, zeroes of 6x^{2} – 3- 7x are 3/2 and 1/3 . Thus, sum of zeroes

Hence, relationship between the zeroes and the coefficients is verified.

(iv) Quadratic polynomial is

Quadratic polynomial is 4u^{2} + 8u = 4u
(u+2)

The value of 4u^{2} +8u is zero, if

if 4u = 0 or u + 2 = 0.

Then u = 0 or u = – 2.

So, zeroes of 4u^{2} + 8u are 0 and
–2.

Thus, sum of zeroes = 0 + (–2) = – 2

Product of zeroes = (0) (–2) = 0

Hence, relationship between the zeroes and the coefficients is verified.

(v) Quadratic polynomial is

The value of t^{2} – 15 is zero if

Hence, relationship between the zeroes and the coefficients is verified.

(vi) Quadratic polynomial is

3x^{2} – x – 4 = 3x^{2} + 3x –
4x – 4

= 3x (x + 1) 4(x + 1)

= (x + 1) (3x – 4)

The value of 3x^{2} – x – 4 is zero, if

(x + 1) = 0 or 3x – 4 = 0

Then x = – 1 or x = 43

So, zeroes of 3x^{2} – x – 4 are –1 and 43.

Hence, the relationship between the zeroes and the coefficients is verified.