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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) \(3\sqrt t+t\sqrt 2\)         

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

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

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

(ii) \(2-x^2+x^3\)               

(iii) \({\pi\over{2}}x^2+x\)

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

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

(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) \(p(x)=lx+m\), \(x=-{m\over l}\)

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

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

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

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