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

Polynomials

Exercise 2.4

cbse class 9 Ncert solutions chapter 2 Polynomials exercise 2.4 question 13
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Class 9

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

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

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

5. Factorise :

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

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

6. Write the following cubes in expanded form:

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

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

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

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

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

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 \(x^3+y^3+z^3-3xyz=\)\({1\over 2}(x+y+z)\)\([(x-y)^2+(y-z)^2+(z-x)^2]\)

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