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14, , EXEMPLAR PROBLEMS, , x3 + y3 = (x + y) (x2 – xy + y2), x3 – y3 = (x – y) (x2 + xy + y2), x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z 2 – xy – yz – zx), , (B) Multiple Choice Questions, Sample Question 1 : 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, Solution : Answer (C), , EXERCISE 2.1, Write the correct answer in each of the following :, 1. Which one of the following is a polynomial?, (A), , x2 2, –, 2 x2, , (B), 3, , (C), , x2 +, , 3x 2, , (D), , x −1, x +1, , (C), , 1, , x, 2., , 2 x −1, , 2 is a polynomial of degree, , (A), , 2, , (B), , 0, , 3. Degree of the polynomial 4x4 + 0x3 + 0x5 + 5x + 7 is, (A) 4, (B) 5, (C) 3, 4. Degree of the zero polynomial is, (A), (D), , 0, (B), Not defined, , 1, , (C), , (D), , 1, 2, , (D), , 7, , Any natural number, , ( ), , 5. If p ( x ) = x2 – 2 2 x + 1 , then p 2 2 is equal to, (A), , 0, , (B), , 1, , (C), , 4 2, 6. The value of the polynomial 5x – 4x2 + 3, when x = –1 is, (A) – 6, (B) 6, (C) 2, , (D), , 8 2 +1, , (D), , –2, , 29052014

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POLYNOMIALS, , 15, , 7. If p(x) = x + 3, then p(x) + p(–x) is equal to, (A) 3, (B) 2x, (C), 8. Zero of the zero polynomial is, (A) 0, (B), (C) Any real number, (D), 9. Zero of the polynomial p(x) = 2x + 5 is, (A), , –, , 2, 5, , (B), , –, , 5, 2, , (C), , 0, , (D), , 6, , (D), , 5, 2, , (D), , –2, , 1, Not defined, 2, 5, , 10. One of the zeroes of the polynomial 2x2 + 7x –4 is, (A), , 2, , (B), , 1, 2, , (C), , –, , 1, 2, , 11. If x51 + 51 is divided by x + 1, the remainder is, (A) 0, (B) 1, (C) 49, (D) 50, 2, 12. If x + 1 is a factor of the polynomial 2x + kx, then the value of k is, (A) –3, (B) 4, (C) 2, (D) –2, 13. x + 1 is a factor of the polynomial, (A) x3 + x2 – x + 1, (B) x3 + x2 + x + 1, (C) x4 + x3 + x2 + 1, (D) x4 + 3x3 + 3x2 + x + 1, 14. One of the factors of (25x2 – 1) + (1 + 5x)2 is, (A) 5 + x, (B) 5 – x, (C) 5x – 1, (D) 10x, 2, 2, 15. The value of 249 – 248 is, (A) 1 2, (B) 477, (C) 487, (D), 2, 16. The factorisation of 4x + 8x + 3 is, (A) (x + 1) (x + 3), (B) (2x + 1) (2x + 3), (C) (2x + 2) (2x + 5), (D) (2x –1) (2x –3), 17. Which of the following is a factor of (x + y)3 – (x3 + y3)?, , 497, , (A) x2 + y2 + 2xy (B) x2 + y2 – xy (C) xy 2, 18. The coefficient of x in the expansion of (x + 3)3 is, (A) 1, (B) 9, (C) 18, , (D), , 3xy, , (D), , 27, , x y, 19. If y + x = –1 ( x , y ≠ 0 ) , the value of x3 – y3 is, , 29052014

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16, , EXEMPLAR PROBLEMS, , (A), , 1, , (B), , –1, , (C), , 0, , (D), , 1, 2, , (D), , 1, 2, , (D), , 2abc, , 1 , 1, , 20. If 49x2 – b =  7 x +   7 x –  , then the value of b is, 2 , 2, , 1, (A), , 0, , (B), , 1, 4, , (C), , 2, , 21. If a + b + c = 0, then a3 + b3 + c3 is equal to, (A) 0, (B) abc, (C) 3abc, , (C) Short Answer Questions with Reasoning, Sample Question 1 : Write whether the following statements are True or False., Justify your answer., 3, , 1, , 1 2, x + 1 is a polynomial, 5, , (i), , 6 x + x2, , (ii), , is a polynomial, x ≠ 0, , x, , Solution :, (i) False, because the exponent of the variable is not a whole number., , (ii), , True, because, , 6 x+, , 3, x2, , = 6 + x , which is a polynomial., , x, , EXERCISE 2.2, 1. Which of the following expressions are polynomials? Justify your answer:, (i), , 8, , (ii), , (iv), , 1, + 5x + 7, 5 x –2, , (v), , (vii), , 1 3 2 2, a –, a + 4a – 7, 7, 3, , (viii), , 3x2 – 2 x, , ( x – 2 )( x – 4 ), x, , (iii), , 1 – 5x, , (vi), , 1, x +1, , 1, 2x, , 29052014

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POLYNOMIALS, , 17, , 2. Write whether the following statements are True or False. Justify your answer., (i) A binomial can have atmost two terms, (ii) Every polynomial is a binomial, (iii) A binomial may have degree 5, (iv) Zero of a polynomial is always 0, (v) A polynomial cannot have more than one zero, (vi) The degree of the sum of two polynomials each of degree 5 is always 5., , (D) Short Answer Questions, Sample Question 1 :, (i) Check whether p(x) is a multiple of g(x) or not, where, p(x) = x3 – x + 1, g(x) = 2 – 3x, (ii) Check whether g(x) is a factor of p(x) or not, where, p(x) = 8x3 – 6x2 – 4x + 3, g(x) =, , x 1, −, 3 4, , Solution :, (i) p(x) will be a multiple of g(x) if g(x) divides p(x)., Now,, , g(x) = 2 – 3x = 0 gives x =, , 2, 3, 3, , Remainder, ,  2 2   2, = p   =   −   +1,  3 3   3, =, , 8 2, 17, − +1 =, 27 3, 27, , Since remainder ≠ 0, so, p(x) is not a multiple of g(x)., (ii) g(x) =, , x 1, 3, − = 0 gives x =, 3 4, 4, ,  3, g(x) will be a factor of p(x) if p   = 0 (Factor theorem),  4, 3, , Now,, , 2, ,  3, 3 , 3 , 3 , p   = 8  − 6   − 4  + 3,  4, 4 , 4 , 4 , , 29052014

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18, , EXEMPLAR PROBLEMS, , = 8×, , Since,, , 27, 9, − 6× −3 +3 = 0, 64, 16, ,  3, p   = 0, so, g(x) is a factor of p(x).,  4, , Sample Question 2 : Find the value of a, if x – a is a factor of x3 – ax2 + 2x + a – 1., Solution : Let p(x) = x3 – ax2 + 2x + a – 1, Since x – a is a factor of p(x), so p(a) = 0., i.e.,, , a 3 – a(a)2 + 2a + a – 1 = 0, a 3 – a3 + 2a + a – 1 = 0, 3a = 1, , Therefore, a =, , 1, 3, , Sample Question 3 : (i)Without actually calculating the cubes, find the value of, 483 – 303 – 183., (ii)Without finding the cubes, factorise (x – y) 3 + (y – z) 3 + (z – x) 3., Solution : We know that x3 + y3 + z 3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)., If x + y + z = 0, then x3 + y3 + z 3 – 3xyz = 0 or x3 + y3 + z 3 = 3xyz., (i) We have to find the value of 483 – 303 – 183 = 483 + (–30)3 + (–18)3., Here, 48 + (–30) + (–18) = 0, So, 483 + (–30)3 + (–18)3 = 3 × 48 × (–30) × (–18) = 77760, (ii) Here, (x – y) + (y – z) + (z – x) = 0, Therefore, (x – y)3 + (y – z) 3 + (z – x) 3 = 3(x – y) (y – z) (z – x)., , EXERCISE 2.3, 1. Classify the following polynomials as polynomials in one variable, two variables etc., (i), (iii), , x2 + x + 1, , (ii), , y3 – 5y, , xy + yz + zx, , (iv), , x2 – 2xy + y2 + 1, , 29052014

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POLYNOMIALS, , 19, , 2. Determine the degree of each of the following polynomials :, (i), , 2x – 1, 3, , (iii) x – 9x + 3x, 3. For the polynomial, , 5, , (ii), , –10, , (iv), , y3 (1 – y4), , x3 + 2 x + 1 7 2, – x – x 6 , write, 5, 2, (i), (ii), (iii), , the degree of the polynomial, the coefficient of x3, the coefficient of x6, , (iv) the constant term, 4. Write the coefficient of x2 in each of the following :, (i), , π, x + x 2 –1, 6, , (ii), , 3x – 5, , (iii) (x –1) (3x –4), (iv) (2x –5) (2x2 – 3x + 1), 5. Classify the following as a constant, linear, quadratic and cubic polynomials :, (i), , 2 – x2 + x3, , (ii), , 3x3, , (iii), , 5t – 7, , (iv), , 4 – 5y2, , (v), , 3, , (vi), , 2+x, , (vii), , y3 – y, , (viii) 1 + x + x2, , (ix), , t2, , (x), , 2x – 1, , 6. Give an example of a polynomial, which is :, (i) monomial of degree 1, (ii) binomial of degree 20, (iii) trinomial of degree 2, 7. Find the value of the polynomial 3x3 – 4x2 + 7x – 5, when x = 3 and also when, x = –3., , 1 , 8. If p(x) = x2 – 4x + 3, evaluate : p(2) – p(–1) + p  , 2 , 9. Find p(0), p(1), p(–2) for the following polynomials :, (i) p(x) = 10x – 4x2 – 3, (ii) p(y) = (y + 2) (y – 2), 10. Verify whether the following are True or False :, (i) –3 is a zero of x – 3, , 29052014

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20, , EXEMPLAR PROBLEMS, , 1, is a zero of 3x + 1, 3, , (ii), , –, , (iii), , –4, is a zero of 4 –5y, 5, , (iv), (v), , 0 and 2 are the zeroes of t2 – 2t, –3 is a zero of y2 + y – 6, , 11. Find the zeroes of the polynomial in each of the following :, (i) p(x) = x – 4, (ii) g(x) = 3 – 6x, (iii) q(x) = 2x –7, (iv) h(y) = 2y, 12. Find the zeroes of the polynomial :, p(x) = (x – 2)2 – (x + 2)2, 13. By actual division, find the quotient and the remainder when the first polynomial is, divided by the second polynomial : x4 + 1; x –1, 14. By Remainder Theorem find the remainder, when p(x) is divided by g(x), where, (i) p(x) = x3 – 2x2 – 4x – 1, g(x) = x + 1, (ii) p(x) = x3 – 3x2 + 4x + 50, g(x) = x – 3, (iii), , p(x) = 4x3 – 12x2 + 14x – 3, g(x) = 2x – 1, , (iv), , p(x) = x3 – 6x2 + 2x – 4, g(x) = 1 –, , 3, x, 2, , 15. Check whether p(x) is a multiple of g(x) or not :, (i) p(x) = x3 – 5x2 + 4x – 3, g(x) = x – 2, (ii) p(x) = 2x3 – 11x2 – 4x + 5, g(x) = 2x + 1, 16. Show that :, (i) x + 3 is a factor of 69 + 11x – x2 + x3 ., (ii) 2x – 3 is a factor of x + 2x3 – 9x2 + 12 ., 17. Determine which of the following polynomials has x – 2 a factor :, (i) 3x2 + 6x – 24, (ii) 4x2 + x – 2, 18. Show that p – 1 is a factor of p 10 – 1 and also of p 11 – 1., 19. For what value of m is x3 – 2mx2 + 16 divisible by x + 2 ?, 20. If x + 2a is a factor of x5 – 4a2 x3 + 2x + 2a + 3, find a., 21. Find the value of m so that 2x – 1 be a factor of 8x4 + 4x3 – 16x2 + 10x + m., , 29052014

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POLYNOMIALS, , 23, , Alternative Solution :, x3 + y3, , = (x + y) 3 – 3xy (x + y), = 123 – 3 × 27 × 12, = 12 [122 – 3 × 27], = 12 × 63 = 756, , EXERCISE 2.4, 1. If the polynomials az3 + 4z2 + 3z – 4 and z3 – 4z + a leave the same remainder, when divided by z – 3, find the value of a., 2. The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by x + 1 leaves, the remainder 19. Find the values of a. Also find the remainder when p(x) is, divided by x + 2., 3. If both x – 2 and x –, , 1, are factors of px 2 + 5x + r, show that p = r., 2, , 4. Without actual division, prove that 2x4 – 5x3 + 2x2 – x + 2 is divisible by x2 – 3x + 2., [ Hint: Factorise x2 – 3x + 2], 5. Simplify (2x – 5y) 3 – (2x + 5y) 3., 6. Multiply x2 + 4y2 + z2 + 2xy + xz – 2yz by (– z + x – 2y)., 7. If a, b, c are all non-zero and a + b + c = 0, prove that, , a 2 b 2 c2, + +, = 3., bc ca ab, , 8. If a + b + c = 5 and ab + bc + ca = 10, then prove that a 3 + b3 + c3 –3abc = – 25., 9. Prove that (a + b + c)3 – a3 – b3 – c 3 = 3(a + b ) (b + c) (c + a)., , 29052014