Page 3.19

### Exercise 3.2

#### Question 1

**Fill up the blanks :**

**(i) When x ^{2}+2x-3 is divided by (x-1) , remainder = __**

Sol :

Divisor=x-1

Zero of x-1 is 1

∴ By remainder theorem ,

Remainder=p(1)=(1)^{2}+2(1)-3

=1+2-3

=0

**(ii) If (x-2) is a factor of the polynomial p(x) , then p(2)= __**

Sol :

⇒By factor theorem : Let p(x) be a polynomial in x of degree n≥1 and let a be any real number , then (x-a) is a factor of p(x) if and only if p(a)=0

⇒It is already given that (x-2) is factor of p(x) then according to theorem , p(2)=0

#### Question 2

**What will be the remainder when y ^{4}-3y^{2}+2y+1 is divided by (y-1) ?**

Sol :

Divisor=y-1

Zero of y-1 is 1

∴ By remainder theorem ,

Remainder=p(1)=(1)^{4}-3(1)^{2}+2(1)+1

=1-3+2+1

=4-3

=1

#### Question 3

**What will be the remainder when x ^{2}+4x+2 is divided by (x+2) ?**

Sol :

Divisor=x+2

Zero of x+2 is -2

∴ By remainder theorem ,

Remainder=p(-2)=(-2)^{2}+4(-2)+2

=4-8+2

=-2

#### Question 4

**If for polynomial p(x) , p(-1)=3 , then what will be the remainder when p(x) is divided by (x+1) ?**

Sol :

=3

#### Question 5

**If for polynomial p(x) , , then write a factor of polynomial p(x)**

Sol :

3x+2

#### Question 6

**If for polynomial p(x), p(-3)=0 , then write a factor of polynomial p(x).**

Sol :

x+3

#### Question 7

**Find the remainder when x ^{3} + 3x^{2} + 3x + 1 is divided by**

**(i) x+1**

Sol :

0

**(ii) **

Sol :

27/8

#### Question 8

**(i) Find the remainder when x ^{3}+1 is divided by x + 1**

Sol :

0

**(ii) Find the remainder when x ^{4}+x^{3}-2x^{2}+x+1 is divided by x-1**

Sol :

2

**(iii) Find the remainder when x3-ax ^{2}+6x-a is divided by x – a.**

Sol :

5a

#### Question 9

**If p(x) = x ^{4} – 3x^{3} + 2x + 1 , then using remainder theorem, find the remainder when p(x) is divided by :**

**(i) x-2**

Sol :

9

**(ii) x-4**

Sol :

217

#### Question 10

**If p(x)=4x ^{3}-3x^{2}+2x-4 , then using remainder theorem, find the remainder when p(x) is divided by :**

**(i) x-1**

Sol :

-1

**(ii) x+1**

Sol :

-13

#### Question 11

**If p(x)=x ^{2}+4x+2 , then what will be the remainder when p(x) is divided by x+2 ?**

Sol :

-2

#### Question 12

**Using factor theorem determine whether x-1 is a factor of the following:**

**(i) x ^{3}+x^{2}-2x+1**

Sol :

No

**(ii) 8x ^{4}-12x^{3}+18x+14**

Sol :

No

**(iii) x ^{3}+8x^{2}-7x-2**

Sol :

Yes

**(iv) **

Sol :

Yes

#### Question 13

**Use factor theorem to determine whether g(x) is a factor of p(x) in each of the following cases :**

**(i) p(x)=2x ^{3}+x^{2}-2x-1 , g(x)=x+1**

Sol :

Yes

**(ii) p(x)=x ^{3}+3x^{2}+3x+1 , g(x)=x+2**

Sol :

No

**(iii) p(x)=x ^{3}-4x^{2}+x+6 , g(x)=x-3**

Sol :

Yes

#### Question 14

**(i) Examine whether 7 + 3x is a factor of 3x ^{3} + 7x.**

Sol :

No

**(ii) Examine whether x+2 is a factor of the polynomials x ^{3}+3x^{2}+5x+6 and 2x+4**

Sol :

Yes

**(iii) Examine whether q(t)=4t ^{3}+4t^{2}-t-1 is a multiple of 2t+1**

Sol :

Yes

#### Question 15

**Using factor theorem, determine whether g(x) is a factor of p(x) in the following pair of polynomials :**

**(i) p(x)=x ^{3}-3x^{2}+2x-12 and g(x)=x-2**

Sol :

No

**(ii) p(x)=x ^{3}+x^{2}+3x+175 and g(x)=x+5**

Sol :

No

**(iii) p(x)=2x ^{3}+4x^{2}-5x-10 and g(x)=x+2**

Sol :

Yes

**(iv) and g(x)=x+√2**

Sol :

Yes

#### Question 16

**Using factor theorem, show that x ^{3}-6x^{2}+11x-6 is divided by x-1**

Sol :

#### Question 17

**(a) In the following polynomials if x- 2 is a factor of each polynomial, then find the value of a in each case :**

**(i) x ^{2}-3x+5a**

Sol :

2/5

**(ii) x ^{3}-2ax^{2}+ax-1**

Sol :

7/6

**(b) If x-1 is a factor of polynomial ax ^{3}-4ax+4a-1 , find the value of a**

Sol :

1

**(c) In the following polynomial if x+a is a factor of each polynomial , find the value of a in each case.**

**(i) x ^{3}+ax^{2}-2x+a+4**

Sol :

-4/3

**(ii) x4-a ^{2}x^{2}+3x-a**

Sol :

0

#### Question 18

**If p(x)=x ^{3}+kx^{2}+hx+6 and x+1 and x-2 are factors of p(x) , then find the value of h and k**

Sol :

h=1 , k=-4

#### Question 19

**If p(x)=x ^{4}-5x^{3}+4x^{2}+ax+b and x-1 and x-2 are factors of p(x) , find the values of a and b**

Sol :

a=8, b=-8

#### Question 20

**If x-1 and x+3 are factors of polynomial f(x)=x ^{2}-hx^{2}-13x+k , find the values of h and k**

Sol :

h=3 , k=15

#### Question 21

**If (x-1) and (x-4) are factors of polynomials p(x)=(x ^{2}-3x+2)(x^{2}+7x+a) and q(x)=(x^{2}+5x+4)(x^{2}-5x+b) , find the values of a and b**

Sol :

a=12 , b=4

#### Question 22

**If f(x)=x ^{2}+px+q , g(x)=x^{2}+lx+m and each is divisible by x+a , then prove that **

Sol :