Polynomials

1

Assignments in Mathematics Class IX (Term I)

2. POLYNOMIALS

IMPOrTANT TerMS, DefINITIONS AND reSuLTS

involved have only non-negative integral powers is called a polynomial. For example, x2 _{+ 5x – 6,}

x3 _{– 7x}2 _{+ 11, x}5 _{– 3x + 2, x}2 _{+ 5, x}4 _{+ 5x}3 _– 2x2 _{+ 7x – 3, etc. are polynomials.}

l In the polynomial 5x3 – 4x2 + 6x – 3, we say that the coefficients of x3_{, x}2 _{and x are 5, – 4 and 6} respectively, and we also say that – 3 is the constant term in it.

l 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

in x of degree 1, 4x2 _– 3

2x – 5 is a polynomial in

x of degree 2, and 3x4 _{– 5x}2 _{+ 1 is a polynomial} in x of degree 4.

l A polynomial of degree 1 is called a linear

polynomial. For example, 3x + 5 is a linear

polynomial in x.

l A polynomial of degree 2 is called a quadratic

polynomial. For example, x2 _{+ 5x –} 1

2 is a quadratic

polynomial in x.

l A polynomial of degree 3 is called a cubic

polynomial. For example, 4x3 _{– 3x}2 _{+ 7x + 1 is a} cubic polynomial in x.

l A polynomial of degree 4 is called a biquadratic

polynomial. For example, x4 _{– 3x}3 _{+ 2x}2 + 5x – 3 is a biquadratic polynomial in x. l A polynomial having one term is called a monomial.

Thus, 5x, 7x2_{, 11x}3_{, 3xy and 2xyz are some examples} of monomials in one, two and three variables. l A polynomial having two terms is called a

binomial. Thus, x + 1, 2x3 _{+ 5, x}2 _{–1, x}6 _{+ 1,}

x + y, x2 _{+ y}2 _{are some examples of binomials in} one and two variables.

l A polynomial having three terms is called a

trinomial. Thus, x2 _{– 3x + 1, x}3 _{– 7x}2 _{+ 11,}

x + y + z are some examples of trinomials.

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

8 , etc. are all constant

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

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

l 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

l The constant polynomial has no zero.

l Every real number is a zero of the zero polynomial.

l A linear polynomial has one and only one zero. l If a polynomial p(x) is divided by d(x) = x – a,

then the remainder is given by p(a).

[degree of p(x) > degree of d(x)]. l 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. l Following results are known as identities as they

are true for all values of the variables a, b and

c. (i) (a + b)2 _{= a}2 _{+ 2ab + b}2 (ii) (a – b)2 _{= a}2 _{– 2ab + b}2 (iii) (a + b) (a – b) = a2 _{– b}2 (iv) (a + b + c)2 _{= a}2 _{+ b}2 _{+ c}2 _{+ 2ab} + 2bc + 2ca (v) (a + b)3 _{= a}3 _{+ b}3 _{+ 3ab (a + b)} (vi) (a – b)3 _{= a}3 _{– b}3 _{– 3ab (a – b)} (vii) a3 _{+ b}3 _{= (a + b)(a}2 _{– ab + b}2₎ (viii) a3 _{– b}3 _{= (a – b)(a}2 _{+ ab + b}2₎ (ix) a3 _{+ b}3 _{+ c}3 _{– 3abc = (a + b + c)} (a2 _{+ b}2 _{+ c}2 _{– ab – bc – ca)}

1. (2x + 5) (2x + 7) is equal to : (a) 4x2 _{+ 12x + 35 (b) 2x}2 _{+ 12x + 35} (c) 4x2 _{+ 24x + 35 (d) 4x}2 _{+ 24x – 35} 2. On factorising x2 _{+ 8x + 15, we get :} (a) (x + 3) (x – 5) (b) (x – 3) (x + 5) (c) (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 ?

(a) x2 _{– 2x – 15 is a multiple of (x – 5)} (b) x2 _{– 2x – 15 is a factor of (x – 5)} (c) (x + 3) is a factor of (x – 5) (d) (x + 3) is a multiple of (x – 5)

4. The value of the polynomial 3x + 2x2 _{– 4 at}

x = 0 is :

(a) 2 (b) 3 (c) – 4 (d) 4 5. x140 _{+ x}139 _{+ x}138 _{+ ... x}2 _{+ x + 1 is a polynomial}

with :

(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 _{– px}2 _{+ 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 _{– py}5 _{+ y}4 _{– py}3 _{+ 3y}

– p – 2, then the value of p is :

(a) 1 (b) 2 (c) 3 (d) – 1 9. On factorising a3+3 3b3, we get : (a)

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12. Which of the following is a monomial of degree 50 ?

(a) 50x (b) x50 _{+ 50} (c) 50x (d) 50x50

13. 8x40 _{+ 3 is a :}

(a) monomial of degree 8 (b) monomial of degree 40 (c) binomial of degree 40 (d) binomial of degree 3

14. If (x – 2) is a factor of the polynomial x4 _{– 2x}3 + ax – 1, then the value of a is :

(a) 1 (b) 0 (c) 1 2 (d) − 1 2 15. If x x − =1 3, then x x 2 2 1 + is : (a) 11 (b) 75 (c) 10 (d) 5 16. The coefficient of x0 _{in 5x}2 _{– 7x – 3 is :} (a) 1 (b) 5 (c) 0 (d) – 3 17. On factorising x2 _{– 3x – 4, we get :} (a) (x – 4) (x + 1) (b) (x – 4) (x – 1) (c) (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 19. If x2 _{+ kx + 6 = (x + 2) (x + 3) for all x, then the}

value of k is :

(a) 1 (b) – 1 (c) 5 (d) 3 20. Which one of the following is a polynomial ?

(a) x x 2 2 2 2 − (b) 2x−1 (c) _x x x 2 3 2 3 + (d) x x − + 1 1

21. Degree of the polynomial 4x4 _{+ 0x}3 _{+ 0x}5 _{+ 5x} + 7 is :

(a) 4 (b) 5 (c) 3 (d) 7

Summative aSSeSSment

Multiple ChoiCe Questions

a. important Questions

TIME -1.5 Hr TEST

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22. If p(x) = x2 _{– 2 2x + 1, then p 2 2}

( )

(a) 0 (b) 1

(c) 4 2 (d) 8 2 1+

23. If x – 1 is a factor of mx2 − 2x+1, then the

value of m is :

(a) 2 (b) 2 1+ (c) 1 (d) 2 1−

24. If x – 1 is a factor of 2x3 _{+ x}2 _{– 4x + m, then the} value of m is : (a) 0 (b) 1 (c) 2 (d) –1 25. On factorising x2 _{+ y}2 _{+ 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 _{+ ax}2 _{– 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 : (a) p(a + b) (b) p b a −    (c) p b a    (d) b_a 28. (x + y)3 _{– (x – y)}3 _{is equal to :} (a) 2(x3 _{+ 3x}2_{y) (b) 2(y}3 _{+ 3x}2_y) (c) 2(x3 _{– 3xy}2_{) (d) 2(y}3 _{– 3x}2_y) 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 _{+ y}4 _{+ x}2_y2_{, we get :} (a) (x2 _{+ y}2 _{+ xy)}2 (b) (x2 _{+ y}2 _{+ xy) (x}2 _{+ y}2 _{– xy)} (c) (1 + x2 _{+ y}2_{) (1 – x}2 _{– y}2₎ (d) (x2 _{– y}2 _{+ xy) (x}2 _{– y}2 _{– xy)} 31. The value of (x – y)3 _{+ (y – z)}3 _{+ (z – x)}3 _{is :} (a) xyz (b) 3xyz (c) (x – y) (y – z) (z – x) (d) 3(x – y) (y – z) (z – x) 32. On factorising x m m x 2₊ ₊ 1 ₊₁   , we get : (a) (x + m) (x – m) (b) x m x m + −    1 ( ) (c) (x m x) m + _ + 1_ (d) (x m x) m − _ − 1_ 33. The coefficient of y in (x + y + z)2 _{is :} (a) 2x (b) 2z (c) x + z (d) 2x + 2z 34. On factorising x a b b a x 2₊ _{+ +1}   ,we get : (a) x a b x a b + −     (b) x a b x b a + +     (c) x a b x b a − −     (d) (x + ab) (x – ab)

35. One of the factors of 2− _{2 2}2

x y is : (a) 1+ 1 xy (b) 1 1 − xy (c) 2 1_ + 1_

xy (d) all the above

36. If ab = 5 and a – b = 2, then the value of a3 _{– b}3 is equal to : (a) 10 (b) 38 (c) – 38 (d) 76 37. If x y z 1 2 1 2 1 2 ₀

+ − = , then the value of (x + y – z)2 is :

(a) 2xy (b) 2yz (c) 4xz (d) 4xy 38. The value of x3 _{+ y}3 _{+ 9xy – 27, if x = 3 – y}

is : (a) 1 (b) – 1 (c) 0 (d) cannot be determined 39. The coefficient of x2 _{in (3x}2 _{+ 2x – 4) (x}2 _{– 3x} – 2) is : (a) 2 (b) – 16 (c) 16 (d) 8 40. The value of x y y z z x x y y z z x − + − + − + +

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41. One of the factors of (25x2 _{– 1) + (1 + 5x)}2 _{is :} (a) 5 + x (b) 5 – x (c) 5x – 1 (d) 10x 42. The value of 2492 _{– 248}2 _{is :} (a) 12 _{(b) 477 (c) 487 (d) 497} 43. The factorisation of 4x2 _{+ 8x + 3 is :} (a) (x + 1) (x + 3) (b) (2x + 1) (2x + 3) (c) (2x + 2) (2x + 5) (d) (2x – 1) (2x – 3) 44. Which of the following is a factor of (x + y)3 _{– (x}3

+ y3_{) ?}

(a) x2 _{+ y}2 _{+ 2xy (b) x}2 _{+ y}2 _{– xy}

(c) xy2 _{(d) 3xy}

45. The coefficient of x in the expansion of (x + 3)3 is : (a) 1 (b) 9 (c) 18 (d) 27 46. If x y y x

+ = −1 (x, y ≠ 0), then the value of x3 _{– y}3 is :

(a) 1 (b) – 1 (c) 0 (d) 1

2

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

(a) 2 (b) 1

2 (c) − 1

2 (d) – 2

48. If x + 1 is a factor of the polynomial 2x2 _{+ kx,} then the value of k is :

(a) – 3 (b) 4 (c) 2 (d) – 2 49. x + 1 is a factor of the polynomial :

(a) x3 _{+ x}2 _{– x + 1} (b) x3 _{+ x}2 _{+ x + 1} (c) x4 _{+ x}3 _{+ x}2 _{+ 1} (d) x4 _{+ 3x}3 _{+ 3x}2 _{+ x + 1} 50. If 49 7 1 2 7 1 2 2

x −b=_ x+ __ x− _, then the value of b is : (a) 0 (b) 1 2 (c) 1 4 (d) 1 2 51. If a + b + c = 0, then a3 _{+ b}3 _{+ c}3 _{is equal to :} (a) 0 (b) abc (c) 3abc (d) 2abc

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)

2. The coefficient of x in the product of (x – 1)(1 – 2x) is : [T-I (2010)] (a) –3 (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 – 3x}2_)(x2 _{– 5) is :}

[T-I (2010)]

(a) –17 (b) –10 (c) –3 (d) 17 5. One of the factors of (x – 1) – (x2 _{– 1) is :}

[T-I (2010)]

(a) x2 _{– 1 (b) x + 1 (c) x – 1 (d) x + 4} 6. The factors of (2a – b)3 _{+ (b – 2c)}3 _{+ 8(c – a)}3

is : [T-I (2010)]

(a) (2a – b)(b – 2c)(c – a)

(b) 3(2a – b)(b – 2c)(c – a) (c) 6(2a – b)(b – 2c)(c – a) (d) 2a × b × 2c

7. In which of the following (x + 2) is a factor ?

[T-I (2010)] (a) 4x3 _{– 13x + 6 (b) x}3 _{+ x}2 _{+ x + 4} (c) 4x3 _{+ 13x – 25 (b) –2x}3 _{+ x}2 _{– x – 19} 8. If P(x) = 7 – 3x + 2x2_{,then value of P(–2) is :} [T-I (2010)] (a) 12 (b) 31 (c) 21 (d) 22 9. If x1/3 _{+ y}1/3 _{+ z}1/3 _{= 0, then which one of the}

following expressions is correct ? [T-I (2010)] (a) x3 _{+ y}3 _{+ z}3 _{= 0}

(b) x + y + z = 3x1/3_y1/3_z1/3 (c) x + y + z = 3xyz (d) x3 _{+ y}3 _{+ z}3 _{= 3xyz}

10. (x + 2) is a factor of 2x3 _{+ 5x}2 _{– x – k. The value}

of k is : [T-I (2010)]

11. The coefficient of x2 _{in (3x + x}3_{) x} x +    1 is : [T-I (2010)] (a) 3 (b) 1 (c) 4 (d) 2 12. What is the remainder when x3 _{– 2x}2 _{+ x + 1 is}

divided by (x – 1)? [T-I (2010)] (a) 0 (b) –1 (c) 1 (d) 2 13. If p x( )= + +2 x x −x , 2 3 2 2 _{then p(–1) is :} [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)]

(a) 1 (b) a (c) 0 (d) 1

a

15. Which of the following is a polynomial in x ?

[T-I (2010)]

(a) x x

+1 (b) x2+ x

(c) x+ 2x2+1 (d) 3x+1

16. The remainder when x2 _{+ 2x + 1 is divided by}

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

18. Which of the following is a binomial in y ?

[T-I (2010)]

(a) y2+ 2 (b) y y

+ +1 2

(c) y+ 2y (d) y y+1

19. The remainder obtained when the polynomial p(x) is divided by (b – ax) is : [T-I (2010)]

(a) p b a −    (b) p a b    (c) p b a    (d) p a b −   

20. Which of the following is a trinomial in x ?

[T-I (2010)] (a) x3 _{+ 1 (b) x}3 _{+ x}2 _{+ x} (c) x x+ x+1 (d) x3 _{+ 2x} 21. a2 _{+ b}2 _{+ c}2 _{– ab – bc – ca equals :} [T-I (2010)] (a) (a + b + c)2 _{(b) (a – b – c)}2 (c) (a – b + c)2 (d) 1 2 2 2 2 [(a b− ) + −(b c) + −(c a) ]

22. If x51 _{+ 51 is divided by (x + 1), the remainder}

is : [T-I (2010)]

(a) 0 (b) 1 (c) 49 (d) 50 23. 2 is a polynomial of degree :

(a) 2 (b) 0 (c) 1 (d) 1

2

24. Which of the following is a polynomial in one

variable ? [T-I (2010)]

(a) 3 – x2 _{+ x (b) 3x+4} (c) x3 _{+ y}3 _{+ 7 (d) x}

x

+1

25. The value of p for which x + p is a factor of

x2 _{+ px + 3 – p is : [T-I (2010)]} (a) 1 (b) –1 (c) 3 (d) –3 26. The degree of the polynomial p(x) = 3 is :

[T-I (2010)] (a) 3 (b) 1 (c) 0 (d) 2 27. If _yx+ = −y_x 1,( ,x y≠0), the value of x3 _{– y}3 _{is :} [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 (3x}2 _{– 5)(4 + 4x}2_{) 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)]

short Answer type Questions

1. Find the remainder when 4x3 _{– 3x}2 _{+ 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 1 2 x + is a polynomial (ii) 6 3 2 x x x + _{is a polynomial, x ≠ 0.}

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

(i) x5 _{– x}4 _{+ 2x}2 _{– 1 (ii) 6 – x}2 (iii) 2x – 5 (iv) 5

4. Find the zeroes of the polynomial p(x) = x2 _{– 5x} + 6.

5. For the polynomial x x x x

3 2 5 2 1 5 7 2 + + − − , write :

(i) the degree of the polynomial

(ii) the coefficient of x3 (iii) the coefficient of x6 (iv) the constant term

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 _{– a}2_x2 _{+ 3x – 6a.}

8. Find the value of the polynomial at the indicated value of variable p x( )=3x2−4x+ 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 _{+ x}2 _{– 2x + 4a – 9,} find the value of a.

12. Verify that 1 is not a zero of the polynomial 4y4 _{– 3y}3 _{+ 2y}2 _{– 5y + 1.}

13. Factorise :

(i) x2 _{+ 9x + 18 (ii) 2x}2 _{– 7x – 15} 14. Expand :

(i) (4a – b + 2c)2 _{(ii) (–x + 2y – 3z)}2 15. Factorise : a3−2 2b3

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 _{+ 20}3 _{– 50}3_{. [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( )=3z2−4z+ 17 when z = 3. [T-I (2010)]

7. Check whether the polynomial t + 1 is a factor of 4t3 _{+ 4t}2 _{– t – 1. [T-I (2010)]} 8. Factorise : x2 x 4 1 8 + − . [T-I (2010)] 9. Factorise : 27 1 216 9 2 1 4 3 2 p − − p + p. [T-I (2010)] 10. If 2x + 3y = 8 and xy = 4, then find the value of

4x2 _{+ 9y}2_{. [T-I (2010)]} 11. If x x 2 2 1 38

+ = , then find the value of

x x −    1 . [T-I (2010)] 12. Check whether the polynomial 3x – 1 is a factor

of 9x3 _{– 3x}2 _{+ 3x – 1. [T-I (2010)]} 13. Find the product of x

x x x x x −     +   +  1 1 2 1 2 , , and x x 4 4 1 +   . [T-I (2010)]

14. Using factor theorem, show that (2x + 1) is a factor of 2x3 _{+ 3x}2 _{– 11x – 6. [T-I (2010)]} 15. Check whether (x + 1) is a factor of x3 _{+ x + x}2

+ 1. [T-I (2010)]

16. Find the value of a if (x – 1) is a factor of

2x2+ax+ 2. _{[T-I (2010)]}

17. Factorise : 7 2x2−10x−4 2. [T-I (2010)] 18. If a + b + c = 7 and ab + bc + ca = 20, find the

19. If – 1 is a zero of the polynomial p(x) = ax3 _{– x}2 + x + 4, find the value of a : [T-I (2010)]

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 : p(x) = 8x3 _{– 6x}2 _{– 4x + 3, g(x) =} x 3 1 4 −

3. Using factor theorem show that x – y is a factor of x (y2 _{– z}2_{) + y (z}2 _{– x}2_{) + z (x}2 _{– y}2_).

4. Find the value of a, if x – a is a factor of x3 _{– ax}2 + 2x + a – 1.

5. Find the value of the polynomial 3x3 _{– 4x}2 _{+ 7x} – 5, when x = 3 and also when x = – 3.

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

short Answer type Questions

a. important Questions

12. If x= −1

3 is a zero of the polynomial p(x) = 27x 3 – ax2 _{– x + 3, then find the value of a.}

[T-I (2010)]

13. Factorise : 64a3 _{– 27b}3 _{– 144a}2_{b + 108ab}2_.

[T-I (2010)] 14. Simplify : (a + b + c)2 _{+ (a – b + c)}2 + (a + b – c)2_{. [T-I (2010)]} 15. If x x +    = 1

9, _{then find the value of} x x 3 3 1 + . [T-I (2010)] 16. Factorise : 4(x2 _{+ 1)}2 _{+ 13(x}2 _{+ 1) – 12.} [T-I (2010)] 17. Factorise : x x x x 2 2 1 2 2 2 + + − − . _{[T-I (2010)]}

18. Determine whether (3x – 2) is a factor of 3x3 _{+ x}2 _{– 20x + 12 ? [T-I (2010)]} 19. Simplify : x− y x y   − +  2 3 2 3 3 3 . [T-I (2010)] 20. Factorise : (2x – y – z)3 _{+ (2y – z – x)}3 _{+ (2z – x} – y)3_{. [T-I (2010)]}

6. Find the zeroes of the polynomial p(x) = (x – 2)2 – (x + 2)2_.

7. What must be added to x3 _{– 3x}2 _{– 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) 999}2

9. Factorise : 16x2 _{+ 4y}2 _{+ 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 _{+ y}2 _{+ 9z}2 _{+ 2xy + 3yz – 6xz).}

1. Find the value of x3 _{+ y}3 _{– 12xy + 64 when}

x + y = –4. [T-I (2010)]

2. If x = 2y + 6, then find the value of x3 _{– 8y}3

– 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 8a3 _{– 8b}3 _{– 36ab}

= 27. [T-I (2010)] 6. If a – b = 7, a2 _{+ b}2 _{= 85, find a}3 _{– b}3_. [T-I (2010)] 7. Expand : (a) 1 ₃ 3 x y +    (b) 4 1 3 3 −   x . [T-I (2010)]

8. The polynomials kx3 _{+ 3x}2 _{– 8 and 3x}3 _{– 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 _{+ 10x}2 _{+ ax + b has (x – 1) and (x + 2) as}

factors. [T-I (2010)]

10. Factorise : 8x3 _{+ y}3 _{+ 27z}3 _{– 18xyz. [T-I (2010)]} 11. If a2 _{+ b}2 _{+ c}2 _{= 90 and a + b + c = 20, then}

find the value of ab + bc + ca. [T-I (2010)]

8 21. If a + b = 11, a2 _{+ b}2 _{= 61, find a}3 _{+ b}3_.

[T-I (2010)]

22. a2+b2+c2=30anda b c+ + =10, then find the value of ab bc ca+ + . [T-I (2010)] 23. Using suitable identity evaluate :

( )423−( )183−( ) .243 [T-I (2010)]

24. Find the values of p and q, if the polynomial x4 ₊

px3 _{+ 2x}2 _{– 3x + q is divisible by the polynomial}

x2 _{– 1. [T-I (2010)]}

25. Simplify (x + y + z)2 _{– (x + y – z)}2_.

[T-I (2010)]

26. Factorise 9x2 _{+ y}2 _{+ z}2 _{– 6xy + 2yz – 6zx. Hence} find its value if x = 1, y = 2 and z = –1.

[T-I (2010)]

27. Find the value of a3 _{+ b}3 _{+ 6ab – 8 when a + b}

= 2. [T-I (2010)]

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

[T-I (2010)]

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)]

30. Find the value of x3 _{+ y}3 _{+ 9xy – 27, if x + y}

= 3. [T-I (2010)]

31. If a + b + c = 6, then find the value of (2 – a)3 + (2 – b)3 _{+ (2 – c)}3 _{– 3(2 – a)(2 – b)(2 – c).} [T-I (2010)] 32. If a2 _{+ b}2 _{+ c}2 _{= 250 and ab + bc + ca = 3,} find a + b + c. [T-I (2010)] 33. If x x

+ =1 7, then find the value of x x 3 3 1 + . [T-I (2010)] 34. If x x

− =1 3, then find the value of x x 3 3 1 − . [T-I (2010)]

35. If ax3 _{+ bx}2 _{+ 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)]

36. Factorise : 2x3 _{– x}2 _{– 13x – 6. [T-I (2010)]} 37. Factorise : a3_{(b – c)}3 _{+ b}3_{(c – a)}3 _{+ c}3_{(a – b)}3_.

[T-I (2010)]

38. If p = 4 – q, prove that p3 _{+ q}3 _{+ 12pq = 64.}

[T-I (2010)]

39. Find the value of k so that 2x – 1 be a factor of 8x4 _{+ 4x}3 _{– 16x}2 _{+ 10x + k. [T-I (2010)]} 40. What are the possible expressions for the dimensions

of the cuboids whose volume is given below ? Volume = 12ky2 _{+ 8ky – 20k. [T-I (2010)]} 41. If the polynomial P(x) = x4 _{– 2x}3 _{+ 3x}2 _{– ax}

+ 8 is divided by (x – 2), it leaves a remainder 10. Find the value of a : [T-I (2010)] 42. Without finding the cubes, find the value of :

1 4 1 3 7 12 3 3 3    +   −   . [T-I (2010)] 43. Simplify : (a + b + c)2 _{– (a – b – c)}2_. [T-I (2010)]

44. Factorise (x – 3y)3 _{+ (3y – 7z)}3 _{+ (7z – x)}3_.

[T-I (2010)] 45. Factorise : 2 2a3+8b3−27c3+18 2abc. [T-I (2010)] 46. Factorise : x6 _{– y}6_{. [T-I (2010)]} 47. If both (x – 2) and x−   1 2 are factors of

px2 _{+ 5x + r, show that p = r. [T-I (2010)]} 48. Find the value of a if (x + a) is a factor of

x4 _{– a}2_x2 _{+ 3x – a. [T-I (2010)]} 49. Factorise by splitting the middle term :

9(x – 2y)2 _{– 4(x – 2y) – 13. [T-I (2010)]} 50. Find the remainder obtained on dividing

2x4−3x3−5x2+ +x 1 by x−1 2.

9 1. If (x + 2) is a factor of x3 _{+ 13x}2 _{+ 32x + 20, then}

factorise it.

2. If the polynomials ax3 _{+ 4x}2 _{+ 3x – 4 and x}3 _{– 4x} + a leave the same remainder when divided by

x – 3, find the value of a.

3. The polynomial p(x) = x4 _{– 2x}3 _{+ 3x}2 _{– 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. 1. Verify : x3 y3 z3 3xy 1 x y z 2 + + − = ( + + ) [(x y− )2+ −(y z)2+ −(z x) ].2 [T-I (2010)] 2. Simplify : ( ) ( ) ( ) ( ) ( ) ( ) . a b b c c a a b b c c a 2 2 3 2 2 3 2 2 3 3 3 3 − + − + − − + − + − [T-I (2010)]

3. Prove that : 2x3 _{+ 2y}3 _{+ 2z}3 _{– 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).}

T-I (2010)]

4. Using factor theorem show that x2 _{+ 5x + 6 is} factor of x4 _{+ 5x}3 _{+ 9x}2 _{+ 15x + 18. [T-I (2010)]} 5. Prove that

(x y z+ + ×) [(x y− )2+ −(y z) ]2 =₂(x3+y3+z3−₃xyz)

[T-I (2010)]

6. The polynomials p(x) = ax3 _{+ 4x}2 _{+ 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)]

7. If both (x + 2) and (2x + 1) are factors of ax2 + 2x + b, prove that a – b = 0. [T-I (2010)] 8. Simplify by factorisation method :

6 2 2 2 2 2 − − − x x x . [T-I (2010)]

9. Show that (x – 1) is a factor of P(x) = 3x3 _{– x}2 – 3x + 1 and hence factorise P(x). [T-I (2010)] 10. The polynomials x3 _{+ 2x}2 _{– 5ax – 8 and x}3 _{+ ax}2 – 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 + y)3 _{– (x – y)}3 _{– 6y(x}2 _{– y}2₎

= 8y3_{. [T-I (2010)]}

long Answer type Questions

a. important Questions

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

5. If a, b, c are all non-zero and a + b + c = 0, prove that a bc b ac c ab 2 2 2 3 + + = . 6. Prove that (a + b + c)3 _{– a}3 _{– b}3 _{– c}3 = 3 (a + b) (b + c) (c + a). 7. If a + b + c = 5 and ab + bc + ca = 10, then prove that a3 _{+ b}3 _{+ c}3 _{– 3abc = – 25.}

B. Questions From CBSE Examination Papers

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)]

13. Simplify by factorisation method : [T-I (2010)]

9 2 3 3 2 2 − − − x x x 14. If p(x) = x3 _{– ax}2 _{+ bx + 3 leaves a remainder} –19 when divided by (x + 2) and a remainder 17 when divided by (x – 2), prove that a + b = 6.

[T-I (2010)]

15. The volume of a cube is given by the polynomial

p(x) = x3 _{– 6x}2 _{+ 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)]

16. Using factor theorem, factorise the polynomial :

x4 _{+ 3x}3 _{+ 2x}2 _{– 3x – 3. [T-I (2010)]} 17. Factorise a7 _{+ ab}6_{. [T-I (2010)]} 18. Using factor theorem, factorise the polynomial.

x4 _{+ 2x}3 _{– 7x}2 _{– 8x + 12. [T-I (2010)]} 19. Without actual division, show that the polynomial

2x4 _{– 5x}3 _{+ 2x}2 _{– x + 2 is exactly divisible by}

x2 _{– 3x + 2. [T-I (2010)]}

20. If x and y be two positive real numbers such that 8x3 _{+ 27y}3 _{= 730 and 2x}2_{y + 3xy}2 _{= 15, then}

evaluate 2x + 3y. [T-I (2010)]

21. Factorise : (x2 _{– 2x)}2 _{– 2(x}2 _{– 2x) – 3.} [T-I (2010)] 22. If x x 2 2 1 51 + = , find (i) x x −1 (ii) x x 3 3 1 − . [T-I (2010)]

23. Find the values of m and n so that the polynomial

f(x) = x3 _{– 6x}2 _{+ mx – n is exactly divisible by} (x – 1) as well as (x – 2). [T-I (2010)] 24. Factorise : x8 _{– y}8_{. [T-I (2010)]} 25. Without actual division prove that x4 _{+ 2x}3 _{– 2x}2

+ 2x – 3 is exactly divisible by x2 _{+ 2x – 3.}

[T-I (2010)]

26. Factorise : a12_x4 _{– a}4_x12_{. [T-I (2010)]} 27. Without actual division, prove that the polynomial

2x4 _{– 5x}3 _{+ 2x}2 _{– x + 2 is exactly divisible by}

x2 _{– 3x + 2. [T-I (2010)]}

28. Factorise : (x2 _{– 3x)}2 _{– 8(x}2 _{– 3x) – 20.}

[T-I (2010)]

29. The polynomial p(x) = x4 _{– 2x}3 _{+ 3x}2 _{– 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 _{+ ax}3 _{– 3x}2 _{+ 2x + b.}

[T-I (2010)]

31. Multiply 9x2 _{+ 25y}2 _{+ 15xy + 12x – 20y + 16 by} 3x – 5y – 4 using suitable identity.

[T-I (2010)]

32. If x2 _{– 3x + 2 is a factor of x}4 _{– ax}2 _{+ b then find}

a and b. [T-I (2010)]

33. Without actual division show that x4 _{+ 2x}3 _{– 2x}2 + 2x – 3 is exactly divisible by x2 _{+ 2x – 3.} [T-I (2010)] 34. Factorise : 27 1 64 27 4 9 16 3 3 2 2 a b a b a b + + + . [T-I (2010)]

35. Find the values of a and b so that (x + 1) and (x – 2) are factors of (x3 _{+ ax}2 _{+ 2x + b).}

[T-I (2010)]

36. Wi 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

(2x4−6x3+3x2+3x−2) is exactly divisible by

(x2−3x+2). [T-I (2010)]

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

– a2_x3 _{+ 2x + a + 3, hence factorise x}2 _{– 2ax – 3.}

[T-I (2010)]

39. The polynomial ax3 _{+ 3x}2 _{– 3 and 2x}3 _{– 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 _{– 4u}2 _{– 12u + 16. [T-I (2010)]} 41. If x x + =1 5, then evaluate x x 6 6 1 + . [T-I (2010)]

42. Without actual division, prove that 2x4 _{– 8x.}3 _{+ 3x}2 + 12x – 9 is exactly divisible by x2 _{– 4x + 3.}

[T-I (2010)]

43. If f(x) = x4 _{– 2x}3 _{+ 3x}2 _{– 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)]