Ncert Solutions for Class 9 Polynomials

Ncert Solutions for Class 9

Polynomials


Polynomials

Exercise 2.1

1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

(i) (4x^2-3x+7)           

(ii) (y^2+sqrt 2)       

(iii) (3sqrt t+tsqrt 2)         

(iv) (y+{2over y})

2. Write the coefficients of (x^2) in each of the following:

(i) (2+x^2+x)

(ii) (2-x^2+x^3)               

(iii) ({piover{2}}x^2+x)

(iv) (sqrt 2x-1)

(v) (x^{10}+y^3+t^{50})

3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.

4. Write the degree of each of the following polynomials:

(i) (5x^3+4x^2+7x)

(ii) (4-y^2)

(iii) (5t-sqrt 7)

(iv) 3

5. Classify the following as linear, quadratic and cubic polynomials:

(i) (x^2+x)

(ii) (x-x^3) 

(iii) (y+y^2+4)

(iv) (1+x)

(v) (3t) 

(vi) (r^2)     

(vii) (7x^3)

Polynomials

Exercise 2.2

1. Find the value of the polynomial (5x-4x^2+3) at

(i) x=0

(ii) x=-1       

(iii) x=2

2. Find p(0), p(1) and p(2) for each of the following polynomials:

(i) (p(y)=y^2-y+1) 

(ii) (p(t)=2+t+2t^2-t^3)

(iii) (p(x)=x^3)

(iv) (p(x)=(x-1)(x+1))

3. Verify whether the following are zeroes of the polynomial, indicated against them.

(i) (p(x)=3x+1), (x=-{1over 3})

(ii) (p(x)=5x-pi), (x={4over 5})               

(iii) (p(x)=x^2-1), (x=1, -1)          

(iv) (p(x)=(x+1)(x-2)), (x=-1, 2)

(v) (p(x)=x^2), (x=0)

(vi) (p(x)=lx+m), (x=-{mover l})

(vii) (p(x)=3x^2-1), (x=-{1over{sqrt 3}},{2over {sqrt 3}})

(viii) (p(x)=2x+1), (x={1over 2})

4. Find the zero of the polynomial in each of the following cases:

(i) p(x) = x + 5

(ii) p(x) = x – 5

(iii) p(x) = 2x + 5

(iv) p(x) = 3x – 2

(v) p(x) = 3x

(vi) p(x) = ax , a ≠ 0

(vii) p(x) = cx + d, c ≠ 0, c, d are real  numbers.          

Polynomials

Exercise 2.3

1. Determine which of the following polynomials has (x + 1) a factor:

(i) (x^3+x^2+x+1)

(ii) (x^4+x^3+x^2+x+1)  

(iii) (x^4+3x^3+3x^2+x+1)  

(iv) (x^3-x^2-(2+sqrt 2)x+sqrt 2)

2. Use the 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)   

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

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

3. Find the value of k, if x – 1 is a factor of p(x) in each of the following cases:

(i) (p(x)=x^2+x+k) 

(ii) (p(x)=2x^2+kx+sqrt 2)

(iii) (p(x)=kx^2-sqrt 2 x+1)

(iv) (p(x)=kx^2-3x+k)

4. Factorise:

(i) (p(x)=12x^2-7x+1)

(ii) (p(x)=2x^2+7x+3)

(iii) (p(x)=6x^2+5x-6)

(iv) (p(x)=3x^2-x-4)

5. Factorise:

(i) (p(x)=x^3-2x^2-x+2)  

(ii) (p(x)=x^3-3x^2-9x-5)

(iii) (p(x)=x^3+13x^2+32x+20)

(iv) (p(y)=2y^3+y^2-2y-1)  

Polynomials

Exercise 2.4

1. Use suitable identities to find the following products:

(i) (x + 4) (x + 10)

(ii) (x + 8) (x – 10)

(iii) (3x + 4) (3x – 5)     

(iv) ((y^2+{3over 2})(y^2-{3over 2}))

(v) ((3-2x)(3+2x))

2. Evaluate the following products without multiplying directly:

(i) 103 × 107

(ii) 95 × 96

(iii) 104 × 96

3. Factorise the following using appropriate identities:

(i) (9x^2+6xy+y^2)

(ii) (4y^2-4y+1)

(iii) (x^2-{y^2over 100})

4. Expand each of the following, using suitable identities:

(i) ((x+2y+4z)^2) 

(ii) ((2x-y+z)^2)

(iii) ((-2x+3y+2z)^2)

(iv) ((3a-7b-c)^2)      

(v) ((-2x+5y-3z)^2)

(vi) ([{1over 4}a-{1over 2}b+1]^2)

5. Factorise :

(i) (4x^2+9y^2+16z^2+12xy-24yz-16xz)

(ii) (2x^2+y^2+8z^2-2sqrt 2 xy+4sqrt 2  yz-8xz) 

6. Write the following cubes in expanded form:

(i) ((2x+1)^3)

(ii) ((2a-3b)^3)

(iii) ([{3over 2}x+1]^3)

(iv) ([x-{2over 3}y]^3)

7. Evaluate the following using suitable identities:

(i) ((99)^3)

(ii) ((102)^3)

(iii) ((998)^3)

8. Factorise each of the following :

(i) (8a^3+b^3+12a^2b+6ab^2)

(ii) (8a^3-b^3-12a^2b+6ab^2)

(iii) (27-125a^3-135a+225a^2)

(iv) (64a^3-27b^3-144a^2b+108ab^2)   

(v) (27p^3-{1over 216}-{9over 2}p^2+{1over 4}p)

9. Verify :

(i) (x^3+y^3=)((x+y)(x^2-xy+y^2))

(ii) (x^3-y^3=)((x-y)(x^2+xy+y^2)) 

10. Factorise each of the following :

(i) (27y^3+125z^3)

(ii) (64m^3-343n^3)

11. Factorise : (27x^3+y^3+z^3-9xyz)

12. Verify that (x^3+y^3+z^3-3xyz=)({1over 2}(x+y+z))([(x-y)^2+(y-z)^2+(z-x)^2])

13. If (x+y+z=0), show that (x^3+y^3+z^3=3xyz).

14. Without actually calculating the cubes, find the value of each of the following:

(i) ((-12)^3+(7)^3+(5)^3)

(ii) ((28)^3+(-15)^3+(-13)^3)

15. Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:

(i) Area : (25a^2-35a+12)

(ii) Area : (35y^2+13y-12)

16. What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

(i) Volume : (3x^2-12x)    

(ii) Volume : (12ky^2+8ky-20k)


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