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KC Sinha Mathematics Solution Class 9 Chapter 3 Polynomials exercise 3.2


Page 3.19

Exercise 3.2

Question 1

Fill up the blanks :

(i) When x2+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 y4-3y2+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 x2+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) , p\left(-\dfrac{2}{3}\right)=0 , 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 x3 + 3x2 + 3x + 1 is divided by

(i) x+1

Sol :

0

(ii) x-\dfrac{1}{2}

Sol :

27/8


Question 8

(i) Find the remainder when x3+1 is divided by x + 1

Sol :

0

(ii) Find the remainder when x4+x3-2x2+x+1 is divided by x-1

Sol :

2

(iii) Find the remainder when x3-ax2+6x-a is divided by x – a.

Sol :

 

5a


Question 9

If p(x) = x4 – 3x3 + 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)=4x3-3x2+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)=x2+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) x3+x2-2x+1

Sol :

No

(ii) 8x4-12x3+18x+14

Sol :

No

(iii) x3+8x2-7x-2

Sol :

Yes

(iv) 2\sqrt{2}x^3+5\sqrt{2}x^2-7\sqrt{2}

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)=2x3+x2-2x-1 , g(x)=x+1

Sol :

Yes

(ii) p(x)=x3+3x2+3x+1 , g(x)=x+2

Sol :

No

(iii) p(x)=x3-4x2+x+6 , g(x)=x-3

Sol :

Yes

 


Question 14

(i) Examine whether 7 + 3x is a factor of 3x3 + 7x.

Sol :

No

(ii) Examine whether x+2 is a factor of the polynomials x3+3x2+5x+6 and 2x+4

Sol :

Yes

(iii) Examine whether q(t)=4t3+4t2-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)=x3-3x2+2x-12 and g(x)=x-2

Sol :

No

(ii) p(x)=x3+x2+3x+175 and g(x)=x+5

Sol :

No

(iii) p(x)=2x3+4x2-5x-10 and g(x)=x+2

Sol :

 

Yes

(iv) p(x)=2\sqrt{2}x^2+5x+\sqrt{2} and g(x)=x+√2

Sol :

 

 

Yes


Question 16

Using factor theorem, show that x3-6x2+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) x2-3x+5a

Sol :

2/5

 

(ii) x3-2ax2+ax-1

Sol :

7/6

 

(b) If x-1 is a factor of polynomial ax3-4ax+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) x3+ax2-2x+a+4

Sol :

-4/3

 

(ii) x4-a2x2+3x-a

Sol :

 

0

 


Question 18

If p(x)=x3+kx2+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)=x4-5x3+4x2+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)=x2-hx2-13x+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)=(x2-3x+2)(x2+7x+a) and q(x)=(x2+5x+4)(x2-5x+b) , find the values of a and b

Sol :

a=12 , b=4


Question 22

If f(x)=x2+px+q , g(x)=x2+lx+m and each is divisible by x+a  , then prove that a=\dfrac{m-q}{l-p}

Sol :

 

 


 

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