CHAPTER 2 POLYNOMIALS (A) Main Concepts and Results • Geometrical meaning of zeroes of a polynomial: The zeroes of a polynomial p(x) are precisely the x-coordinates of the points where the graph of y = p(x) intersects the x-axis. • Relation between the zeroes and coefficients of a polynomial: If α and β are the bc –zeroes of a quadratic polynomial ax2 + bx + c, then α + β , αβ . aa • If α, β and γ are the zeroes of a cubic polynomial ax3 + bx2 + cx + d, then b c–d –α+β+γ , α β + β γ + γ α and α β γ . a aa • The division algorithm states that given any polynomial p(x) and any non-zero polynomial g( x), there are polynomials q(x) and r(x) such that p(x) = g(x) q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x). (B) Multiple Choice Questions Choose the correct answer from the given four options: Sample Question 1: If one zero of the quadratic polynomial x2 + 3x + k is 2, then the value of k is (A) 10 (B) –10 (C) 5 (D) –5 Solution : Answer (B) Sample Question 2: Given that two of the zeroes of the cubic polynomial ax3 + bx2 + cx + d are 0, the third zero is –bbc d (A) (B) (C) (D) – aaa a Solution : Answer (A). [Hint: Because if third zero is α, sum of the zeroes –b = α + 0 + 0 = ]a EXERCISE 2.1 Choose the correct answer from the given four options in the following questions: 1. If one of the zeroes of the quadratic polynomial (k–1) x2 + k x + 1 is –3, then the value of k is 4 –42 –2(A) (B) (C) (D)333 3 2. A quadratic polynomial, whose zeroes are –3 and 4, is (A) x2 – x + 12 (B) x2 + x + 12 2xx(C) 2 –2–6 (D) 2x2 + 2x –24 3. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and –3, then (A) a = –7, b = –1 (B) a = 5, b = –1 (C) a = 2, b = – 6 (D) a = 0, b = – 6 4. The number of polynomials having zeroes as –2 and 5 is (A)1 (B) 2 (C) 3 (D) more than 3 5. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is cc b (A) – (B) (C) 0 (D) – aa a 6. If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the product of the other two zeroes is (A) b – a + 1 (B) b – a – 1 (C) a – b + 1 (D) a – b –1 7. The zeroes of the quadratic polynomial x2 + 99x + 127 are (A) both positive (B) both negative (C) one positive and one negative (D) both equal 8. The zeroes of the quadratic polynomial x2 + kx + k, k ≠ 0, (A) cannot both be positive (B) cannot both be negative (C) are always unequal (D) are always equal 9. If the zeroes of the quadratic polynomial ax2 + bx + c, c ≠ 0 are equal, then (A) c and a have opposite signs (B) c and b have opposite signs (C) c and a have the same sign (D) c and b have the same sign 10. If one of the zeroes of a quadratic polynomial of the formx2+ax + b is the negative of the other, then it (A) has no linear term and the constant term is negative. (B) has no linear term and the constant term is positive. (C) can have a linear term but the constant term is negative. (D) can have a linear term but the constant term is positive. 11. Which of the following is not the graph of a quadratic polynomial? (A)(B) (C)(D) (C) ShortAnswer Questions with Reasoning Sample Question 1: Can x – 1 be the remainder on division of a polynomial p (x) by 2x + 3? Justify your answer. Solution : No, since degree (x – 1) = 1 = degree (2x + 3). Sample Question 2: Is the following statement True or False? Justify your answer. If the zeroes of a quadratic polynomial ax2 + bx + c are both negative, then a, b and c all have the same sign. b bSolution : True, because – = sum of the zeroes < 0, so that > 0. Also the product aacof the zeroes = > 0. aEXERCISE 2.2 1. Answer the following and justify: (i) Can x2 – 1 be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5? (ii) What will the quotient and remainder be on division of ax2 + bx + c by px3 + qx2 + rx + s, p ≠ 0? (iii) If on division of a polynomial p (x) by a polynomial g (x), the quotient is zero, what is the relation between the degrees of p (x) and g (x)? (iv) If on division of a non-zero polynomial p (x) by a polynomial g (x), the remainder is zero, what is the relation between the degrees of p (x) and g (x)? (v) Can the quadratic polynomial x2 + kx + k have equal zeroes for some odd integer k > 1? 2. Are the following statements ‘True’ or ‘False’? Justify your answers. (i) If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign. (ii) If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial. (iii) If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial. (iv) If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms. (v) If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign. (vi) If all three zeroes of a cubic polynomial x3 + ax2 – bx + c are positive, then at least one of a, b and c is non-negative. (vii) The only value of k for which the quadratic polynomial kx2 + x + k has 1 equal zeros is 2 (D) Short Answer Questions 1Sample Question 1:Find the zeroes of the polynomial x2 + x – 2, and verify the6 relation between the coefficients and the zeroes of the polynomial. 11 1 Solution : x2 + x – 2 = 6 (6x2 + x – 12) = 6 [6x2 + 9x – 8x – 12]6 1 1 = [3x (2x + 3) – 4 (2x + 3)] = (3x – 4) (2x + 3)6 643 Hence, and – are the zeroes of the given polynomial.321The given polynomial is x2 + x – 2.6 4 3 –1 Coefficient of x –The sum of zeroes = 3+ = – 2 and2 6 Coefficient of x 4 –3 Constant term –2the product of zeroes = = 23 2Coefficient of x EXERCISE 2.3 Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: 1. 4x2 – 3x – 1 2. 3x2 + 4x – 4 3. 5t2 + 12t + 7 4. t3 – 2t2 – 15t 73 5. 2x2 + x + 6. 4x2 + 52 x – 324 7. 2s2 – (1 + 22)s + 2 8. v2 + 43 v – 15 3 11 2 9. y2 + 5 y – 5 10. 7y2 – y – 2 33 (E) Long Answer Questions Sample Question 1: Find a quadratic polynomial, the sum and product of whose 3 zeroes are 2 and – 2 , respectively. Also find its zeroes. Solution : A quadratic polynomial, the sum and product of whose zeroes are 3 3 –2 and – is x2 – 2 x22 31 x2 – 2 x – = [2x2 – 22 x – 3]221 = [2x2 + 2 x – 32x – 3]21 = [2 x (2 x + 1) – 3 ( 2 x + 1)]21 = [2 x + 1] [ 2 x – 3]213 Hence, the zeroes are – and 2.2Sample Question 2: If the remainder on division of x3 + 2x2 + kx +3 by x – 3 is 21, find the quotient and the value of k. Hence, find the zeroes of the cubic polynomial x3 + 2x2 + kx – 18. Solution : Let p(x) = x3 + 2x2 + kx + 3 Then, p(3) = 33 + 2 × 32 + 3k + 3 = 21 i.e., 3k = –27 i.e., k = –9 Hence, the given polynomial will become x3 + 2x2 – 9x + 3. Now, x – 3) x3 + 2x2 – 9x +3(x2 + 5x +6 x3 – 3x2 5x2 – 9x +3 5x2 – 15x 6x + 3 6x – 18 21 So, x3 + 2x2 – 9x + 3 = (x2 + 5x + 6) (x – 3) + 21 i.e., x3 + 2x2 – 9x – 18 = (x – 3) (x2 + 5x + 6) = (x – 3) (x + 2) (x + 3) So, the zeroes of x3 +2x2 +kx –18 are 3, – 2, – 3. EXERCISE 2.4 1. For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation. –84 215 (i) 3, (ii) 8,3 16 –3 1 (iii) –23,–9 (iv) 25, – 2 2. Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial. 3. Given that 2 is a zero of the cubic polynomial 6x3 + 2 x2 – 10x – 4 2, find its other two zeroes. 4. Find k so that x2 + 2x + k is a factor of 2x4 + x3 – 14 x2 + 5x + 6. Also find all the zeroes of the two polynomials. 5. Given that x – 5 is a factor of the cubic polynomial x3 – 3 5x2 + 13x – 3 5, find all the zeroes of the polynomial. 6. For which values of a and b, are the zeroes of q(x) = x3 + 2x2 + a also the zeroes of the polynomial p(x) = x5 – x4 – 4x3 + 3x2 + 3x + b? Which zeroes of p(x) are not the zeroes of q(x)?