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Cynesure Institute POLYNOMIAL Class 10 POLYNOMIAL CLASS 10 1. A quadratic polynomial whose product and sum of zeroes are 1/5 and 3 respectively. 3x2 – 32x +1. 2. Check if x2 + 4 is factor of x4 + 9x2 + 20. 3. Find a and b if the zeroes of q(x) = x3 + 2x2 + a are also the zeroes of the polynomial p(x) = x5 – x4 – 4x3 + 3x2 + 3x + b? 4. Find a quadratic polynomial, the sum and product of whose zeroes are √2 and – 3/2 respectively. 1 5. Find a quadratic polynomial, the sum and the product of whose zeroes are and –2 2 respectively. 6. Find all the three zeroes of 2x3 – 11x2 + 17x – 6. 7. Find k if one zero of the quadratic polynomial x2 + 3x + k is 2. 8. Find the value of k for which - 4 is zero of the polynomial x2 – x – (2k+2)? 9 9. Find the value of k such that x2 + 2x + k is a factor of 2x4 + x3 – 14 x2 + 5x + 6. Then find all the zeroes of the two polynomials. 10. Find the zeroes of the polynomial 2x2 - 3x - 2. 11. Find the zeroes of the polynomials 2x2 – (1 + 2√ 2)x + √2and verify the relations between the zeroes and the coefficients of the polynomials. 12. Find the zeroes of the polynomials 4x2 + 5 √2x – 3 and verify the relations between the zeroes and the coefficients of the polynomials. 2 y – and verify the relations between the 3 3 zeroes and the coefficients of the polynomials. 13. Find the zeroes of the polynomials 7y2 – 11 Downloaded from cynesure.com 14. Find the zeroes of the polynomials y2 + 2 √5y – 5 and verify the relations between the 3 zeroes and the coefficients of the polynomials. 15. Find the zeroes of the quadratic polynomial 2x2 - 9 - 3x and verify the relationship between the zeroes and the coefficients. Ans: 3, -3/2 16. For what values of m and n are x = 2 and –3 zeroes of the polynomial: 3x2 – 2mx + 2n. 17. Form a quadratic polynomial, the sum and product of whose zeroes are 3 and -2. x2-3x-2 18. Form the polynomial whose zeroes are 2 +3 and 2 - 3. Ans:p(x)=x2-4x+1 19. How many polynomials can have zeroes –2 and 5? 2 20. If , are zeroes of p(x)=kx2+4x+4 such that 2 +2=24, find k. Ans: k=3 or k= -1 21. If , are zeroes of the polynomial x2-2x-15, then find a quadratic polynomial whose zeroes are 2 and 2. Ans: x2-4x-60 22. If , are zeroes of the quadratic polynomial ax2+bx+c, then find (a) + (b) 2 +2 (c) 2 - 2 (d) 3 +3 23. If √2 is a zero of the cubic polynomial 6x3 + √2 x2 – 10x – 4√2, find its other two zeroes. 24. If 1 is a zero of p(x) = ax2 -3(a-1)x -1, find the value of a. 1 25. If 2 is a zero of the cubic polynomial 6x3 + 2 x2 – 10x – 4 √2 , find other two zeroes. 26. If one zero of p(x) = (k–1) x2 + k x + 1 is (–3), find k. 27. If one zero of the cubic polynomial p(x)= x3 + ax2 + bx + c is (–1), find the product of the other two zeroes. 28. If the square of difference of the zeroes of p(x)=x2+px+45 is144, find the value of p. 29. If two zeroes of the cubic polynomial ax3 + bx2 + cx + d are 0, find the third zero. Downloaded from cynesure.com 30. If x – √5 is a factor of the cubic polynomial x3 – 3√ 5 x2 + 13x – 3√5 , find all the zeroes of the polynomial. 31. If zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are a, a + b, a + 2b, find the values of a and b. 32. If zeros of quadratic polynomial x2 + (a + 1) x + b are –3 and 2, find a and b. 33. If α and β . are zeroes of the polynomial 2x2 + 7x - 3, find the value of α2 + β2 . 61/4 34. It is given that 2 is a zero of both the polynomials 3x2+ax-14 and 2x3+bx2+x-2, then find the value of 2-2b. 9 35. What are the sign of the zeroes of the quadratic polynomial x2 + 99x + 127? 36. If P(x)= 2x2-3x+5,3x+5, find p(-2). 37. Find the zeroes of P(x) = x2-2x-3. 38. If a and b are the zeros of 2x2+5x-10, find ab and a+ b. 39. Find the value of ‘k’ such that the quadratic polynomial x2-(k+6) x+2(2k+1) has sum of the zeros is half of their product. 40. If a and b are the zeroes of the quadratic polynomial p(x) = x2-p(x+1)-c, show that (a +1) (b +1)=1-c. 41. If the sum of the zeroes of the quadratic polynomial p(t) = kt2+2t+3k is equal to their product, find the value of k. 42. Find the zeroes of P(x) = 2x2+9x-35. 43. Form the quadratic polynomial whore zeros are 3 and -5. 44. If a and b are the zeros of the quadratic polynomial p(x) = x2-px+q, find a2 + b2. 45. Find k so that the zeroes of the quadratic polynomial 3x2-kx+14 are in the ratio 7:6. 46. If a zero of the quadratic polynomial p(x) = 4x2-8kx-9 is negative of the other zero, find the value of k. 47. If a and b are the zeros of the polynomial f(x) = x2+px+q form polynomial whore zeros are (a + b )2 and (a - b )2. Downloaded from cynesure.com 1. Find the remaining zeroes of the polynomial 2x3 + x2 - 6x - 3, if two of its zeroes are -3 and 3. x=-1/2 2. Find the remaining zeroes of the polynomial p(x)=2x4 +7x3-19x2-14x +30 if two of its zeroes are 2 and -2. Ans: 3/2 and -5 3. Find the remaining zeroes of x4 – 3x3 – 5x2 + 21x – 14 if two of its zeroes are √7 and – √7. 4. Divide the polynomial (x4 + 1) by (x – 1). Also verify the division algorithm. 5. On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4 respectively, find g(x). 6. Find p if the polynomial 6x3+16x2+px -5 is exactly divisible by 3x+5. 7 7. If p(x) = x4+2x3+8x2+12x+18 is divided by x2+5, the remainder comes out to be px+q. Find p and q. p=2,q=3 8. Divide 30x4 +11x3-82x2-12x-48 by (3x2 +2x-4). 9. Obtain the remaining zeroes of 3x4 -15x3+13x2+25x-30, if two of its zeroes are 5 3 5 3 and - . 2, 3 10. What must be added to 6x5+5x4+11x3-3x2+x+5 so that it may be exactly divisible by 3x22x+4? 11. Check if x – 1 can be the remainder on division of a polynomial by 2x + 3? 12. Find quotient and remainder on division of ax2 + bx + c by px3 + qx2 + rx + s? 13. If p (x) is divided by a polynomial g (x), the quotient is zero, what is the relation between p (x) and g (x)? 14. If remainder when x3 + 2x2 + kx +3 is divided by x – 3 is 21, find the quotient and the value of k. 15. Divide x4-5x+6 by 1- x2. 16. Cheek whether the polynomial t2-3 is a factor of the polynomial 2t4+3t3-2t2-9t-12 . Downloaded from cynesure.com Downloaded from cynesure.com Downloaded from cynesure.com