## Ex 2.1

### Question 1:

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

Solution:

## Ex 2.2

### Question 1.

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

**(i) x**^{2} – 2x – 8

^{2}– 2x – 8

### x^{2} – 4x +2x – 8

### x (x-4) 2 (x-4)

### (x-4)(x+2)

### x-4 = 0

**x = 4**

### x+2 = 0

**x = -2**

### So zeroes are 4 and -2

**Verifications:**

### x^{2} – 4x +2x – 8

### sum of zeroes = -b/a

### 4+(-2) = -(-2)/1

### 4-2 = 2/1

### 2 =2 verify

### product of zeroes = c/a

### 4 x -2 = -8/1

### -8 = -8 verify

**(ii) 4s**^{2} – 4s + 1

^{2}– 4s + 1

### 4s^{2} – 2s - 2s + 1

### 2s(2s-1) -1 (2s-1)

### (2s-1)(2s-1)

### 2s-1 = 0

### 2s = 1

**s = 1/2**

### so, zeroes are 1/2 , 1/2

**Verifications:**

### 4s^{2} – 4s + 1

### sum of zeroes = -b/a

### 1/2 + 1/2 = -(-4)/4

### 2/2 = 4/4

### 1 =1 verify

### product of zeroes = c/a

### 1/2 x 1/2 = 1/4

### 1/4 = 1/4 verify

** (iii) 6x**^{2} – 3 – 7x

^{2}– 3 – 7x

### first we write in order

### 6x^{2} - 7x - 3

6x^{2} - 9x + 2x -3

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

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

### 2x - 3 = 0

### 2x = 3

**x = 3/2**

### 3x + 1 = 0

### 3x = -1

**x = -1/3**

### so, zeroes are 3/2 and -1/3

**Verifications:**

### 6x^{2} - 7x - 3

### sum of zeroes = -b/a

### 3/2 +(- 1/3) = -(-7)/6

### 3/2 - 1/3 = 7/6

### 7/6 = 7/6 verify

### product of zeroes = c/a

### 3/2 x -1/3 = -3/6

### -3/6 = -3/6 verify

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

^{2}+ 8u

### 4u is common so,

### 4u ( u + 2)

### 4u = 0

**u = 0**

### u + 2 = 0

**u = -2**

**Verifications:**

### 4u^{2} + 8u

### sum of zeroes = -b/a

### 0 + ( -2) = - 8/4

### -2 = -2 verify

### product of zeroes = c/a

### 0 x -2 = 0/4

### 0 = 0 verify

**(v) t**^{2} – 15

^{2}– 15

## **t**^{2} – ()^{2 } . : X = 15

^{2}– ()

^{2 }. : X = 15

From Identitity A^{2} - B ^{2} = ( A + B ) (A - B)

**(t** + **) ****(t** - **)**

**)**

**)**

### S0, zeroes are ^{ }and - ^{}

^{ }and -^{}**Verifications:**

### sum of zeroes = -b/a

^{ }-^{ } ^{ } = 0 = - 0/1

^{ }-

^{ }= 0 = - 0/1### product of zeroes = c/a

^{ x }^{ - } ^{ = - 15 = -15/1 verify}

^{ x }

^{ = - 15 = -15/1 verify}**(vi) 3x**^{2} – x – 4

^{2}– x – 4

### 3x^{2} + 3x - 4 x – 4

### 3x ( x+1) -4 (x+1)

### (3x - 4) ( x + 1)

### (3x - 4) = 0

** x = 4/3**

### ( x + 1) = 0

** x = -1**

### S0, zeroes are 4/3 and -1

**Verifications:**

### sum of zeroes = -b/a

### 4/3 + (-1) = -(-1)/3

### 4/3 - 1 = 1/3 = 1/3

### product of zeroes = c/a

### 4/3 x -1 = -4/3 verify

### Question 2.

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

## Ex 2.3

### Question 1.

Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:

(i) p(x) = x^{3} – 3x^{2} + 5x – 3, g(x) = x^{2} – 2

(ii) p(x) = x^{4} – 3x^{2} + 4x + 5, g(x) = x^{2} + 1 – x

(iii) p(x) = x^{4}– 5x + 6, g(x) = 2 – x^{2
Solution:}