Polynomials

Assignments in Mathematics Class IX (Term I) 2. POLYNOMIALS IMPOrTANT TerMS, DefINITIONS AND reSuLTS l

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polynomials. In general, every real number is a constant polynomial. Clearly, the degree of a nonzero constant polynomial is zero.

An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial. For example, x2 + 5x – 6, x3 – 7x2 + 11, x5 – 3x + 2, x2 + 5 , x4 + 5x3 – 2x2 + 7x – 3, etc. are polynomials. In the polynomial 5x3 – 4x2 + 6x – 3, we say that the coefficients of x3, x2 and x are 5, – 4 and 6 respectively, and we also say that – 3 is the constant term in it.



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A polynomial consisting of one term namely zero only, is called a zero polynomial. The degree of zero polynomial is not defined.

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Let p(x) be a polynomial. If p(α) = 0, then we say that α is a zero of the polynomial p(x). Finding the zeroes of a polynomial p(x) means solving the equation p(x) = 0

In case of a polynomial in one variable, the highest power of the variable is called the degree of the polynomial. For example, 2x + 3 is a polynomial 3 in x of degree 1, 4x2 – x – 5 is a polynomial in 2 x of degree 2, and 3x4 – 5x2 + 1 is a polynomial in x of degree 4.

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The constant polynomial has no zero.

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Every real number is a zero of the zero polynomial.

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A linear polynomial has one and only one zero.

A polynomial of degree 1 is called a linear polynomial. For example, 3x + 5 is a linear polynomial in x.

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If a polynomial p(x) is divided by d(x) = x – a, then the remainder is given by p(a).

A polynomial of degree 2 is called a quadratic 1 polynomial. For example, x2 + 5x – is a quadratic 2 polynomial in x.

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[degree of p(x) > degree of d(x)]. factor Theorem : Let f(x) be a polynomial of degree n > 1 and let a be any real number. (i) If f(a) = 0, then (x – a) is a factor of f(x). (ii) If (x – a) is a factor of f(x), then f(a) = 0.

A polynomial of degree 3 is called a cubic polynomial. For example, 4x3 – 3x2 + 7x + 1 is a cubic polynomial in x.

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A polynomial of degree 4 is called a biquadratic polynomial. For example, x 4 – 3x 3 + 2x 2 + 5x – 3 is a biquadratic polynomial in x.

Following results are known as identities as they are true for all values of the variables a, b and c. (i) (a + b)2 = a2 + 2ab + b2 (ii) (a – b)2 = a2 – 2ab + b2

A polynomial having one term is called a monomial. Thus, 5x, 7x2, 11x3, 3xy and 2xyz are some examples of monomials in one, two and three variables.

(iii) (a + b) (a – b) = a2 – b2 (iv) (a + b + c)2 = a2 + b2 + c2 + 2ab

A polynomial having two terms is called a binomial. Thus, x + 1, 2x3 + 5, x2 –1, x6 + 1, x + y, x2 + y2 are some examples of binomials in one and two variables.

+ 2bc + 2ca (v) (a + b)3 = a3 + b3 + 3ab (a + b) (vi) (a – b)3 = a3 – b3 – 3ab (a – b)

A polynomial having three terms is called a trinomial. Thus, x2 – 3x + 1, x3 – 7x2 + 11, x + y + z are some examples of trinomials.

(vii) a3 + b3 = (a + b) (a2 – ab + b2) (viii) a3 – b3 = (a – b) (a2 + ab + b2) (ix) a 3 + b 3 + c 3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)

A polynomial containing one term only, consisting of a constant is called a constant polynomial. 7 , etc. are all constant For example, 3, – 5, 8

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TIME -1.5 Hr

TEST

Summative aSSeSSment

Multiple ChoiCe Questions

[1 Mark]

a. important Questions 1. (2x + 5) (2x + 7) is equal to : 4x2

(a) (x – 1) (x – 2) (x – 3)

2x2

(a) + 12x + 35 (c) 4x2 + 24x + 35

(b) + 12x + 35 (d) 4x2 + 24x – 35

2. On factorising x2 + 8x + 15, we get : (a) (x + 3) (x – 5) (c) (x + 3) (x + 5)

(b) (x – 3) (x + 5) (d) (x – 3) (x – 5)

3. On dividing x2 – 2x – 15 by (x – 5), the quotient is (x + 3) and remainder is 0. Which of the following statements is true ?

(c) (x + 1) (x – 1) (x – 2) (d) (x + 1) (x – 1) (x + 2) 12. Which of the following is a monomial of degree 50 ? (a) 50x

(b) x50 + 50

(c)

(d)

50x

(b) x2 – 2x – 15 is a factor of (x – 5)

(a) monomial of degree 8

(c) (x + 3) is a factor of (x – 5)

(b) monomial of degree 40 (c) binomial of degree 40

(d) (x + 3) is a multiple of (x – 5) 4. The value of the polynomial 3x + x = 0 is : (a) 2 x140

+ with :

(b) 3 x139

+

x138

(c) – 4 + .........

x2

2x2

– 4 at

(d) 4

+ x + 1 is a polynomial

(a) infinite terms

(b) 140 terms

(c) 141 terms

(d) 139 terms

6. The coefficient of x in (x + 5) (x – 7) is : (a) – 12

(b) 2

(c) – 2

(d) 12

7. The remainder when x3 – px2 + 6x – p is divided by x – p is : (a) p

(b) 5p

(c) – 5p

(d) 5p2

8. If (y – p) is a factor of y6 – py5 + y4 – py3 + 3y – p – 2, then the value of p is : (a) 1

(b) 2

(c) 3

(d) – 1

9. On factorising a 3 + 3 3b3 , we get : (a) (b) (c) (d)

(a + 3b)(a

2

(a + 3b)(a2 − (a + 3b)(a2 − ( 3a + b)(a2 − (b) 1

14. If (x – 2) is a factor of the polynomial x4 – 2x3 + ax – 1, then the value of a is : (a) 1

(b) 0

15. If x −

1 x

(a) 11

(c) 2

(d) −

2

= 3 , then x + 2 is : x (b) 75 (c) 10 x0

16. The coefficient of (a) 1

1

1

(b) 5

17. On factorising

x2

in

5x2

1 2

(d) 5

– 7x – 3 is :

(c) 0

(d) – 3

– 3x – 4, we get :

(a) (x – 4) (x + 1) (c) (x + 4) (x – 1)

(b) (x – 4) (x – 1) (d) (x + 4) (x + 1)

18. If p(x) = x + 3, then p(x) + p(–x) is equal to : (a) 3

(b) 2x

(c) 0

(d) 6

2

19. If + kx + 6 = (x + 2) (x + 3) for all x, then the value of k is :

2

20. Which one of the following is a polynomial ?

(a) 1

(a)

2

2

(c) 840

(b) – 1

x2

2 − 2 2 x

(c) x 2 +

10. The value of 73 + 83 – 153 is : (a) 0

(d) binomial of degree 3

x2

) 3ab + 3b ) 3ab − 3b ) 3ab + 3b )

+ 3ab + 3b

50 x 50

13. 8x40 + 3 is a :

(a) x2 – 2x – 15 is a multiple of (x – 5)

5.

(b) (x + 1) (x – 2) (x – 3)

(d) – 2520

11. On factorising x3 – 2x2 – x + 2, we get :

3 3x 2

x

(c) 5

(b)

(d)

(d) 3

2x − 1 x −1 x +1

21. Degree of the polynomial 4x4 + 0x3 + 0x5 + 5x + 7 is : (a) 4 (b) 5 (c) 3 (d) 7

( )

22. If p(x) = x2 – 2 2x + 1, then p 2 2 is equal to : (a) 0 (b) 1 (d) 8 2 + 1

(c) 4 2

23. If x – 1 is a factor of mx 2 − 2 x + 1 , then the value of m is : (a)

2 +1

(b)

2

24. If x – 1 is a factor of value of m is : (a) 0

(b) 1

(c) 1

2x3

+

x2

(d)

2 −1

– 4x + m, then the

(c) 2

(d) –1

25. On factorising x2 + y2 + 2 (xy + yz + zx), we get : (a) (x + y) (x + y + z) (b) (x + y + z)2 (c) (x + y) (x + y + 2z) (d) (x + y) (x + y + z)2 26. The value of a for which (x + a) is a factor of x3 + ax2 – 3x + 16 + a is : (a) – 4

(b) 4

(c) – 2

(d) 2

27. When p(x) is divided by ax – b, then the remainder is :

 −b   a

(b) p  

(a) p(a + b)

 b (c) p    a 28. (x +

y)3

– (x –

y)3

a

is equal to : (b) 2(y3 + 3x2y)

(c) 2(x3 – 3xy2)

(d) 2(y3 – 3x2y)

+

 

29. If p(x) = q(x) × g(x) + r(x), r(x) ≠ 0, where p(x), q(x), g(x), and r(x) are polynomials, then : (a) degree of r(x) = degree of g(x) (b) degree of r(x) > degree of g(x) (c) degree of r(x) < degree of g(x) (d) degree of r(x) = 0 30. On factorising x4 + y4 + x2y2, we get :

(a) 2x

(b) 2z

(c) x + z

(d) 2x + 2z

a   (a)  x +   x −  b  a   (c)  x −   x −  b 

b +  x + 1, we get : b a a  b  (b)  x +   x +   a b 

a

 b b

(d) (x + ab) (x – ab)



a

2 is : 35. One of the factors of 2 − 2 2 x y (a) 1 +

1

(b) 1 −

xy

 

2 1 +

xy

1

(a) 10 1

(b) 38 1

(c) – 38

(d) 76

1

37. If x 2 + y 2 − z 2 = 0, then the value of (x + y – z)2 is : (a) 2xy

(b) 2yz x3

38. The value of is :

+

(c) 4xz y3

(d) 4xy

+ 9xy – 27, if x = 3 – y

(a) 1

(d) cannot be determined

(d) 3(x – y) (y – z) (z – x)

1

(d) all the above xy  36. If ab = 5 and a – b = 2, then the value of a3 – b3 is equal to : (c)

(c) (1 + x2 + y2) (1 – x2 – y2)

(c) (x – y) (y – z) (z – x)



a

34. On factorising x 2 +  

(c) 0

(b) 3xyz

1

m

33. The coefficient of y in (x + y + z)2 is :

(b) (x2 + y2 + xy) (x2 + y2 – xy)

(a) xyz

 ( x − m)

m

 

(b) – 1

31. The value of (x – y)3 + (y – z)3 + (z – x)3 is :

1

(d) ( x − m)  x −



m

(a) (x2 + y2 + xy)2

(d) (x2 – y2 + xy) (x2 – y2 – xy)

 x + 1, we get :

m

(b)  x + 

1

(c) ( x + m)  x +

1



(a) (x + m) (x – m)

b

3x2y)

(a)

2(x3

(d)

 

32. On factorising x 2 +  m +

39. The coefficient of x2 in (3x2 + 2x – 4) (x2 – 3x – 2) is : (a) 2 40. The value of is :

(b) – 16

(x

(c) 16

( x − y )3 + ( y − z )3 + ( z − x)3 2

−

y2

) + (y 3

(a) (x – y) (y – z) (z – x) 1 (b) ( x − y )( y − z )( z − x ) (c)

(d) 8

1

( x + y )( y + z )( z + x )

(d) (x + y) (y + z) (z + x)

2

−

z2

) + (z 3

2

−

x2

)

3

(a) 1

(b) – 1

(c) 0

(d)

1

2 47. One of the zeroes of the polynomial 2x2 + 7x – 4 is :

41. One of the factors of (25x2 – 1) + (1 + 5x)2 is : (a) 5 + x (b) 5 – x 42. The value of (a) 12

2492

–

(c) 5x – 1 2482

(b) 477

(d) 10x

is :

(c) 487

(a) (c)

xy2

+

y2

(b) (2x + 1) (2x + 3) (d) (2x – 1) (2x – 3)

+ 2xy

(b)

x2

+

y2

– xy

45. The coefficient of x in the expansion of (x + 3)3 is :

46. If

x y

(b) 9 y

+

x

(c) 18

(a) – 3

(b) 4

(c) 2

(d) – 2

(b) x3 + x2 + x + 1

(d) 3xy

(a) 1

1

(a) x3 + x2 – x + 1

44. Which of the following is a factor of (x + y)3 – (x3 + y3) ? x2

(c) −

(d) – 2 2 2 48. If x + 1 is a factor of the polynomial 2x2 + kx, then the value of k is : (b)

49. x + 1 is a factor of the polynomial :

(d) 497

43. The factorisation of 4x2 + 8x + 3 is : (a) (x + 1) (x + 3) (c) (2x + 2) (2x + 5)

1

(a) 2

(d) 27

= −1 (x, y ≠ 0), then the value of x3 – y3

(c) x4 + x3 + x2 + 1 (d) x4 + 3x3 + 3x2 + x + 1 1   50. If 49 x 2 − b =  7 x +   7 x −  2  of b is : 1

(a) 0

(b)

(a) 0

(b) abc

(c)

1

 , then the value

2

1

(d)

1

4 2 2 3 3 3 51. If a + b + c = 0, then a + b + c is equal to : (c) 3abc

(d) 2abc

is :

B. Questions From CBSE Examination Papers 1. If a polynomial f(x) is divided by x – a, then remainder is : [T-I (2010)] (a) f(0)

(b) f(a)

(c) f(–a)

(d) f(a) – f(0)

(b) 3

(c) –2

(d) 1

3. One of the factors of (x3 – 1) – (x – 1) is : [T-I (2010)] (a) x + 1 (b) x2 – 1

(c) x – 1

(d) x + 4

4. The coefficient of x2 in (2 – 3x2)(x2 – 5) is : [T-I (2010)] (a) –17

(b) –10

(c) –3

5. One of the factors of (x – 1) –

(d) 17 (x2

(a)

– 1 (b) x + 1

– 1) is :

6. The factors of (2a – is :

(c) x – 1 b)3

+ (b –

(a) (2a – b)(b – 2c)(c – a)

2c)3

7. In which of the following (x + 2) is a factor ? [T-I (2010)] (a)

4x3

(c)

4x3

– 13x + 6 + 13x – 25

(b)

x3

(b)

–2x3

+

x2

+x+4

+ x2 – x – 19

8. If P(x) = 7 – 3x + 2x2, then value of P(–2) is : [T-I (2010)] (a) 12 x1/3

(b) 31 y1/3

(c) 21

(d) 22

z1/3

9. If + + = 0, then which one of the following expressions is correct ? [T-I (2010)] (a) x3 + y3 + z3 = 0

[T-I (2010)] x2

(c) 6(2a – b)(b – 2c)(c – a) (d) 2a × b × 2c

2. The coefficient of x in the product of (x – 1)(1 – 2x) is : [T-I (2010)] (a) –3

(b) 3(2a – b)(b – 2c)(c – a)

(d) x + 4 a)3

+ 8(c – [T-I (2010)]

(b) x + y + z = 3x1/3y1/3z1/3 (c) x + y + z = 3xyz (d) x3 + y3 + z3 = 3xyz 10. (x + 2) is a factor of 2x3 + 5x2 – x – k. The value of k is : [T-I (2010)] (a) 6

(b) –24

(c) –6

(d) 24

1  11. The coefficient of x2 in (3x + x3)  x +  is :  x [T-I (2010)] (a) 3 (b) 1 (c) 4 (d) 2 12. What is the remainder when x3 – 2x2 + x + 1 is divided by (x – 1)? [T-I (2010)] (a) 0 (b) –1 (c) 1 (d) 2 x x2 2 13. If p ( x) = 2 + + x − , then p(–1) is : 2 3 [T-I (2010)] (a)

15 6

(b)

17 6

(c)

1 6

(d)

13 6

14. Zero of the polynomial p(x) where p(x) = ax, a ≠ 0 is : [T-I (2010)] 1 (a) 1 (b) a (c) 0 (d) a 15. Which of the following is a polynomial in x ? [T-I (2010)] (a) x +

1 x

(b) x 2 + x

(d) 3 x + 1 (c) x + 2 x 2 + 1 2 16. The remainder when x + 2x + 1 is divided by (x + 1) is : [T-I (2010)] (a) 4 (b) 0 (c) 1 (d) –2 1  1  2 1   17. Product of  x −   x +   x + 2  is :  x  x  x  [T-I (2010)] (a) x 4 −

1 x4

(c) x 4 −

1

(b) x 3 + (d) x 2 +

1

−2

x3 1

+2 x x2 18. Which of the following is a binomial in y ? [T-I (2010)] 1 (a) y 2 + 2 (b) y + + 2 y 4

y + 2y

(c)

(d) y y + 1

19. The remainder obtained when the polynomial p(x) is divided by (b – ax) is : [T-I (2010)]  −b   a (b) p   (a) p    a  b  b (c) p    a

 −a  (d) p    b

20. Which of the following is a trinomial in x ? [T-I (2010)] (a)

x3

+1

(c) x x + x + 1

(b)

x3

+

x2

(d) x3 + 2x

+x

21. a2 + b2 + c2 – ab – bc – ca equals : [T-I (2010)] (a) (a + b +

c)2

(b) (a – b – c)2

(c) (a – b + c)2 1 [(a − b)2 + (b − c)2 + (c − a)2 ] 2 22. If x51 + 51 is divided by (x + 1), the remainder is : [T-I (2010)] (d)

(a) 0 23.

(b) 1

(c) 49

(d) 50

2 is a polynomial of degree :

1 2 24. Which of the following is a polynomial in one variable ? [T-I (2010)] (a) 2

(b) 0

(a) 3 – x2 + x

(c) 1

(b)

(d)

3x + 4

1 x 25. The value of p for which x + p is a factor of x2 + px + 3 – p is : [T-I (2010)] (c) x3 + y3 + 7

(d) x +

(a) 1

(c) 3

(b) –1

(d) –3

26. The degree of the polynomial p(x) = 3 is : [T-I (2010)] (a) 3 (b) 1 (c) 0 (d) 2 x y 27. If + = −1,( x, y ≠ 0), the value of x3 – y3 is : y x [T-I (2010)] (a) 1 (b) –1 (c) 1/2 (d) 0 28. (1 + 3x)3 is an example of : [T-I (2010)] (a) monomial (b) binomial (c) trinomial (d) none of these 29. Degree of zero polynomial is : [T-I (2010)] (a) 0 (b) 1 (c) any natural number (d) not defined 30. The coefficient of x2 in (3x2 – 5)(4 + 4x2) is : [T-I (2010)] (a) 12 (b) 5 (c) –8 (d) 8 31. One of the factors of (16y2 – 1) + (1 – 4y)2 is : [T-I (2010)] (a) (4 + y) (b) (4 – y) (c) (4y + 1) (d) 8y 32. If x2 + kx + 6 = (x + 2)(x + 3) for all x, the value of k is : [T-I (2010)] (a) 1 (b) –1 (c) 5 (d) 3 33. Zero of the zero polynomial is : [T-I (2010)] (a) 0 (b) 1 (c) any real number (d) not defined 34. If (x – 1) is a factor of p(x) = x2 + x + k, then value of k is : [T-I (2010)] (a) 3

(b) 2

(c) –2

(d) 1

short Answer type Questions

[2 Marks]

a. important Questions 1. Find the remainder when 4x3 – 3x2 + 2x – 4 is divided by x + 2. 2. Write whether the following statements are true or false : In each case justify your answer. (i)

1 5

1

x 2 + 1 is a polynomial 3

(ii)

6 x+x 2 x

is a polynomial, x ≠ 0.

3. Write the degree of each of the following polynomials : (i) x5 – x4 + 2x2 – 1

(ii) 6 – x2

(iv) 5 5 4. Find the zeroes of the polynomial p(x) = x2 – 5x + 6. x3 + 2 x + 1 7 2 5. For the polynomial − x − x5 , 5 2 write : (iii) 2x –

6. Give an example of a polynomial which is : (i) monomial of degree 1 (ii) binomial of degree 20 7. Find the value of a, if x + a is a factor of the polynomial x4 – a2x2 + 3x – 6a. 8. Find the value of the polynomial at the indicated value of variable p ( x ) = 3 x 2 − 4 x + 11, at x = 2. 9. Find p(1), p(–2) for the polynomial p(x) = (x + 2) (x – 2). 10. Show that x + 3 is a factor of 69 + 11x – x2 + x3. 11. If (x + 1) is a factor of ax3 + x2 – 2x + 4a – 9, find the value of a. 12. Verify that 1 is not a zero of the polynomial 4y4 – 3y3 + 2y2 – 5y + 1. 13. Factorise : (i) x2 + 9x + 18

(ii) 2x2 – 7x – 15

14. Expand :

(i) the degree of the polynomial

(i) (4a – b + 2c)2

(ii) the coefficient of x3

(ii) (–x + 2y – 3z)2

15. Factorise : a 3 − 2 2b3

(iii) the coefficient of x6 (iv) the constant term

B. Questions From CBSE Examination Papers 1. Evaluate using suitable identity (999)3. [T-I (2010)] 2. Factorise : 3x2 – x – 4.

[T-I (2010)]

3. Using factor theorem, show that (x + 1) is a factor of x19 + 1. [T-I (2010)] 4. Without actually calculating the cubes, find the value of 303 + 203 – 503. [T-I (2010)] 5. Evaluae (104)3 using suitable identity. [T-I (2010)] 6. F i n d t h e v a l u e o f t h e p o l y n o m i a l p ( z ) = 3 z 2 − 4 z + 17 when z = 3. [T-I (2010)] 7. Check whether the polynomial t + 1 is a factor of 4t3 + 4t2 – t – 1. [T-I (2010)] x 1 8. Factorise : x 2 + − . 4 8 [T-I (2010)] 9. Factorise : 27 p3 −

1 9 1 − p2 + p. 216 2 4 [T-I (2010)]

10. If 2x + 3y = 8 and xy = 4, then find the value of 4x2 + 9y2. [T-I (2010)]

11. If x 2 +

1 x2

= 38, then find the value of

1   x −  . x

[T-I (2010)]

12. Check whether the polynomial 3x – 1 is a factor of 9x3 – 3x2 + 3x – 1. [T-I (2010)] 1  1  1  13. Find the product of  x −  ,  x +  ,  x 2 + 2   x  x  x   4 1 and  x + 4  . [T-I (2010)]  x  14. Using factor theorem, show that (2x + 1) is a factor of 2x3 + 3x2 – 11x – 6. [T-I (2010)] 15. Check whether (x + 1) is a factor of x3 + x + x2 + 1. [T-I (2010)] 16. Find the value of a if (x – 1) is a factor of 2 x 2 + ax + 2 . 17. Factorise : 7 2 x 2 − 10 x − 4 2 .

[T-I (2010)] [T-I (2010)]

18. If a + b + c = 7 and ab + bc + ca = 20, find the [T-I (2010)] value of a2 + b2 + c2.

19. If – 1 is a zero of the polynomial p(x) = ax3 – x2 + x + 4, find the value of a : [T-I (2010)]

20. Check whether the polynomial p(s) = 3s3 + s2 – 20s + 12 is a multiple of 3s – 2. [T-I (2010)] 21. Factorise : 125x3 + 27y3.

[T-I (2010)]

short Answer type Questions

[3 Marks]

a. important Questions 1. Check whether p(x) is a multiple of g(x) or not, where p(x) = x3 – x + 1, g(x) = 2 – 3x. 2. Check whether g(x) is a factor of p(x) or not, where : x 1 p(x) = 8x3 – 6x2 – 4x + 3, g(x) = − 3 4 3. Using factor theorem show that x – y is a factor of x (y2 – z2) + y (z2 – x2) + z (x2 – y2). 4. Find the value of a, if x – a is a factor of x3 – ax2 + 2x + a – 1. 5. Find the value of the polynomial 3x3 – 4x2 + 7x – 5, when x = 3 and also when x = – 3.

6. Find the zeroes of the polynomial p(x) = (x – 2)2 – (x + 2)2. 7. What must be added to x3 – 3x2 – 12x + 19 so that the result is exactly divisible by x2 + x – 6 ? 8. Using suitable identity, evaluate the following : (i) 1033

(ii) 101 × 102

(iii) 9992

9. Factorise : 16x 2 + 4y2 + 9z 2 – 16xy – 12yz + 24xz 10. If x + y + z = 9 and xy + yz + zx = 26, find x2 + y2 + z 2. 11. Find the following product : (2x – y + 3z) (4x2 + y2 + 9z2 + 2xy + 3yz – 6xz).

B. Questions From CBSE Examination Papers 1. Find the value of x3 + y3 – 12xy + 64 when x + y = –4. [T-I (2010)] 2. If x = 2y + 6, then find the value of x3 – 8y3 – 36xy – 216. [T-I (2010)] 3. Factorise : 27(x + y)3 – 8(x – y)3.

[T-I (2010)]

4. Factorise : (x – 2y)3 + (2y – 3z)3 + (3z – x)3. [T-I (2010)] 5. If 2a = 3 + 2b, prove that = 27.

8a3

–

8b3

– 36ab [T-I (2010)]

6. If a – b = 7, a2 + b2 = 85, find a3 – b3.

−1 is a zero of the polynomial p(x) = 27x3 3 – ax2 – x + 3, then find the value of a.

12. If x =

[T-I (2010)] 13. Factorise : 64a3 – 27b3 – 144a2b + 108ab2. [T-I (2010)] c) 2

14. Simplify : (a + b + + (a – b + c) 2 2 + (a + b – c) . [T-I (2010)] 1  1 15. If  x +  = 9, then find the value of x 3 + 3 .   x x

[T-I (2010)]  1 y 7. Expand : (a)  +   x 3

3

3

1  (b)  4 −  .  3x  [T-I (2010)]

8. The polynomials kx3 + 3x2 – 8 and 3x3 – 5x + k are divided by x + 2. If the remainder in each case is the same, find the value of k. [T-I (2010)] 9. Find the values of a and b so that the polynomial x3 + 10x2 + ax + b has (x – 1) and (x + 2) as factors. [T-I (2010)] 10. Factorise : 8x3 + y3 + 27z3 – 18xyz. [T-I (2010)] 11. If a2 + b2 + c2 = 90 and a + b + c = 20, then find the value of ab + bc + ca. [T-I (2010)]

[T-I (2010)] 16. Factorise :

4(x2

+

1)2

+

13(x2

+ 1) – 12. [T-I (2010)]

17. Factorise : x 2 +

1

2 + 2 − 2x − . x x

[T-I (2010)]

2

18. Determine whether (3x – 2) is a factor of [T-I (2010)] 3x3 + x2 – 20x + 12 ? 3

3

2  2    19. Simplify :  x − y −  x + y .   3  3 

[T-I (2010)]

20. Factorise : (2x – y – z)3 + (2y – z – x)3 + (2z – x – y)3. [T-I (2010)]

36. Factorise : 2x3 – x2 – 13x – 6.

21. If a + b = 11, a2 + b2 = 61, find a3 + b3. [T-I (2010)]

37. Factorise : a3(b – c)3 + b3(c – a)3 + c3(a – b)3.

22. a 2 + b 2 + c 2 = 30 and a + b + c = 10, then find the value of ab + bc + ca. [T-I (2010)] 23. Using suitable identity evaluate : (42)3 − (18)3 − (24)3 .

[T-I (2010)] 38. If p = 4 – q, prove that p3 + q3 + 12pq = 64.

[T-I (2010)]

[T-I (2010)]

24. Find the values of p and q, if the polynomial x4 + px3 + 2x2 – 3x + q is divisible by the polynomial x2 – 1. [T-I (2010)]

39. Find the value of k so that 2x – 1 be a factor of 8x4 + 4x3 – 16x2 + 10x + k. [T-I (2010)]

25. Simplify (x + y + z)2 – (x + y – z)2.

40. What are the possible expressions for the dimensions of the cuboids whose volume is given below ?

[T-I (2010)]

Volume = 12ky2 + 8ky – 20k.

26. Factorise 9x2 + y2 + z2 – 6xy + 2yz – 6zx. Hence find its value if x = 1, y = 2 and z = –1. 27. Find the value of = 2.

+

b3

+ 6ab – 8 when a + b [T-I (2010)]

28. If x + y + z = 9, then find the value of (3 – + (3 – y)3 + (3 – z)3 – 3(3 – x)(3 – y)(3 – z).

42. Without finding the cubes, find the value of : 3

x)3

3

3

 1  1  7   +   −   . 4 3 12

[T-I (2010)]

[T-I (2010)] 44. Factorise (x – 3y)3 + (3y – 7z)3 + (7z – x)3.

30. Find the value of x3 + y3 + 9xy – 27, if x + y = 3. [T-I (2010)]

[T-I (2010)]

a)3

31. If a + b + c = 6, then find the value of (2 – + (2 – b)3 + (2 – c)3 – 3(2 – a)(2 – b)(2 – c).

45. Factorise : 2 2 a3 + 8b3 − 27c3 + 18 2 abc. [T-I (2010)]

[T-I (2010)] 46. Factorise : x6 – y6.

32. If a2 + b2 + c2 = 250 and ab + bc + ca = 3, find a + b + c. [T-I (2010)]

[T-I (2010)]

1  47. If both (x – 2) and  x −  are factors of  2 2 px + 5x + r, show that p = r. [T-I (2010)]

1 1 3 = 7, then find the value of x + 3 . x x

48. Find the value of a if (x + a) is a factor of x4 – a2x2 + 3x – a. [T-I (2010)]

[T-I (2010)] 34. If x −

[T-I (2010)]

43. Simplify : (a + b + c)2 – (a – b – c)2.

29. If x – 3 is a factor of x2 – kx + 12, then find the value of k. Also, find the other factor for this value of k. [T-I (2010)]

33. If x +

[T-I (2010)]

41. If the polynomial P(x) = x4 – 2x3 + 3x2 – ax + 8 is divided by (x – 2), it leaves a remainder 10. Find the value of a : [T-I (2010)]

[T-I (2010)] a3

[T-I (2010)]

1 1 3 = 3, then find the value of x − 3 . x x

49. Factorise by splitting the middle term : 9(x – 2y)2 – 4(x – 2y) – 13.

[T-I (2010)]

[T-I (2010)]

50. Find the remainder obtained on dividing 1 2 x 4 − 3 x 3 − 5 x 2 + x + 1 by x − . 2 [T-I (2010)]

35. If ax3 + bx2 + x – 6 has (x + 2) as a factor and leaves a remainder 4 when divided by x – 2, find the values of a and b. [T-I (2010)]

8

long Answer type Questions

[4 Marks]

a. important Questions 1. If (x + 2) is a factor of factorise it.

x3

+

13x2

4. Without actual division prove that (x – 2) is a factor of the polynomial 3x3 – 13x2 + 8x + 12. Also, factorise it completely.

+ 32x + 20, then

5. If a, b, c are all non-zero and a + b + c = 0, prove a 2 b2 c2 + + = 3. that bc ac ab 6. Prove that (a + b + c) 3 – a 3 – b 3 – c 3 = 3 (a + b) (b + c) (c + a).

2. If the polynomials ax3 + 4x2 + 3x – 4 and x3 – 4x + a leave the same remainder when divided by x – 3, find the value of a. 3. The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by x + 1, leaves the remainder 19. Find the value of a. Also, find the remainder when p(x) is divided by x + 2.

7. If a + b + c = 5 and ab + bc + ca = 10, then prove that a3 + b3 + c3 – 3abc = – 25.

B. Questions From CBSE Examination Papers 1 ( x + y + z) 2 [( x − y)2 + ( y − z )2 + ( z − x )2 ]. [T-I (2010)]

12. Find the value of (x – a)3 + (x – b)3 + (x – c)3 – 3(x – a)(x – b)(x – c), if a + b + c = 3x. [T-I (2010)]

1. Verify : x 3 + y3 + z 3 − 3 xy =

13. Simplify by factorisation method :

2 2 3 2 2 3 2 2 3 2. Simplify : (a − b ) + (b − c ) + (c − a ) . 3 3 3 ( a − b) + (b − c ) + ( c − a)

9 − 2 3x − x

3 − x2 – + bx + 3 leaves a remainder 14. If p(x) = –19 when divided by (x + 2) and a remainder 17 when divided by (x – 2), prove that a + b = 6.

[T-I (2010)]

x3

3. Prove that : 2x3 + 2y3 + 2z3 – 6xyz = (x + y + z) [(x – y)2 + (y – z)2 + (z – x)2]. Hence evaluate 2(7)3 + 2(9)3 + 2(13)3 – 6(7) (9) (13).

15. The volume of a cube is given by the polynomial p(x) = x3 – 6x2 + 12x – 8. Find the possible expressions for the sides of the cube. Verify the truth of your answer when the length of cube is 3 cm. [T-I (2010)]

4. Using factor theorem show that x2 + 5x + 6 is factor of x4 + 5x3 + 9x2 + 15x + 18. [T-I (2010)] 5. Prove that ( x + y + z ) × [( x − y)2 + ( y − z)2 ] = 2( x 3 + y3 + z 3 − 3 xyz )

16. Using factor theorem, factorise the polynomial : x4 + 3x3 + 2x2 – 3x – 3. [T-I (2010)]

[T-I (2010)] 6. The polynomials p(x) = ax3 + 4x2 + 3x – 4 and q(x) = x3 – 4x + a leave the same remainder when divided by x – 3. Find the remainder when p(x) is divided by (x – 2). [T-I (2010)]

17. Factorise a7 + ab6. x4 + 2x3 – 7x2 – 8x + 12.

8. Simplify by factorisation method : 6 − 2 2x − x2 [T-I (2010)] . 2 − x2 9. Show that (x – 1) is a factor of P(x) = 3x3 – x2 – 3x + 1 and hence factorise P(x). [T-I (2010)]

20. If x and y be two positive real numbers such that 8x3 + 27y3 = 730 and 2x2y + 3xy2 = 15, then evaluate 2x + 3y. [T-I (2010)] 21. Factorise : (x2 – 2x)2 – 2(x2 – 2x) – 3.

10. The polynomials x3 + 2x2 – 5ax – 8 and x3 + ax2 – 12x – 6 when divided by (x – 2) and (x – 3) leave remaindens p and q respectively. If q – p = 10, find the value of a. [T-I (2010)] 11. Prove that (x + = 8y3.

– (x –

–

[T-I (2010)]

19. Without actual division, show that the polynomial 2x4 – 5x3 + 2x2 – x + 2 is exactly divisible by x2 – 3x + 2. [T-I (2010)]

7. If both (x + 2) and (2x + 1) are factors of + 2x + b, prove that a – b = 0. [T-I (2010)]

6y(x2

[T-I (2010)]

18. Using factor theorem, factorise the polynomial.

ax2

y)3

ax2

[T-I (2010)]

T-I (2010)]

y)3

[T-I (2010)]

2

22. If x 2 +

1

x2 1 (i) x − x

y 2)

– [T-I (2010)]

9

[T-I (2010)] = 51, find (ii) x 3 −

1 x3

.

[T-I (2010)]

23. Find the values of m and n so that the polynomial f(x) = x3 – 6x2 + mx – n is exactly divisible by (x – 1) as well as (x – 2). [T-I (2010)] 24. Factorise : x8 – y8.

[T-I (2010)] x4

25. Without actual division prove that + 2x3 – 2x2 2 + 2x – 3 is exactly divisible by x + 2x – 3. [T-I (2010)] 26. Factorise : a12x4 – a4x12.

[T-I (2010)]

27. Without actual division, prove that the polynomial 2x4 – 5x3 + 2x2 – x + 2 is exactly divisible by x2 – 3x + 2. [T-I (2010)] 28. Factorise :

(x2

–

3x)2

–

8(x2

– 3x) – 20. [T-I (2010)]

29. The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by (x + 1), leaves the remainder 19. Find the value of a. Also, find the remainder, when p(x) is divided by x + 2. [T-I (2010)] 30. Find the values of a and b so that (x + 1) and (x – 1) are factors of x4 + ax3 – 3x2 + 2x + b. [T-I (2010)] 31. Multiply 9x2 + 25y2 + 15xy + 12x – 20y + 16 by 3x – 5y – 4 using suitable identity. [T-I (2010)] 32. If x2 – 3x + 2 is a factor of x4 – ax2 + b then find a and b. [T-I (2010)] 33. Without actual division show that x4 + 2x3 – 2x2 + 2x – 3 is exactly divisible by x2 + 2x – 3. [T-I (2010)]

27a2 9a + . 3 4b 16b2 64b [T-I (2010)] 35. Find the values of a and b so that (x + 1) and (x – 2) are factors of (x3 + ax2 + 2x + b). [T-I (2010)]

34. Factorise : 27a3 +

1

+

36. W i t h o u t a c t u a l d i v i s i o n , p r o v e t h a t (2 x 4 − 6 x 3 + 3 x 2 + 3 x − 2) is exactly divisible by [T-I (2010)] ( x 2 − 3 x + 2). 37. Simplify : (5a + 3b)3 – (5a – 3b)3. [T-I (2010)] 38. Find the value of a if (x – a) is a factor of x5 – a2x3 + 2x + a + 3, hence factorise x2 – 2ax – 3. [T-I (2010)] 39. The polynomial ax3 + 3x2 – 3 and 2x3 – 5x + a when divided by x – 4 leave the same remainder in each case. Find the value of a. [T-I (2010)] 40. Factorise : 3u3 – 4u2 – 12u + 16. [T-I (2010)] 1 1 41. If x + = 5 , then evaluate x 6 + 6 . x x [T-I (2010)] 42. Without actual division, prove that 2x4 – 8x.3 + 3x2 + 12x – 9 is exactly divisible by x2 – 4x + 3. [T-I (2010)] 43. If f(x) = x4 – 2x3 + 3x2 – ax + b is divided by (x – 1) and (x + 1), it leaves the remainders 5 and 19 respectively. Find a and b. [T-I (2010)]