Page 1 X – Maths 9 CHAPTER 2 POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively. 2. A quadratic polynomial in x with real coefficient is of the form ax 2 + bx + c, where a, b, c are real number with a ? 0. 3. 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 i.e. x = a is a zero of polynomial p(x) if p(a) = 0. 4. A polynomial can have at most the same number of zeros as the degree of polynomial. 5. For quadratic polynomial ax 2 + bx + c (a ? 0) Sum of zeros ? ? b a Product of zeros ? . c a 6. The division algorithm states that given any polynomial p(x) and polynomial g(x), there are polynomials q(x) and r(x) such that : p(x) = g(x).q (x) + r(x), g(x) ? 0 where r(x) = 0 or degree of r(x) < degree of g(x). MULTIPLE CHOICE QUESTIONS 1. A real no. ? is a zero of the polynomial f(x) if (a) f( ?) > 0 (b) f( ?) = 0 (c) f( ?) < 0 (d) none Page 2 X – Maths 9 CHAPTER 2 POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively. 2. A quadratic polynomial in x with real coefficient is of the form ax 2 + bx + c, where a, b, c are real number with a ? 0. 3. 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 i.e. x = a is a zero of polynomial p(x) if p(a) = 0. 4. A polynomial can have at most the same number of zeros as the degree of polynomial. 5. For quadratic polynomial ax 2 + bx + c (a ? 0) Sum of zeros ? ? b a Product of zeros ? . c a 6. The division algorithm states that given any polynomial p(x) and polynomial g(x), there are polynomials q(x) and r(x) such that : p(x) = g(x).q (x) + r(x), g(x) ? 0 where r(x) = 0 or degree of r(x) < degree of g(x). MULTIPLE CHOICE QUESTIONS 1. A real no. ? is a zero of the polynomial f(x) if (a) f( ?) > 0 (b) f( ?) = 0 (c) f( ?) < 0 (d) none 10 X – Maths 2. The zeros of a polynomial f(x) are the coordinates of the points where the graph of y = f(x) intersects (a) x-axis (b) y-axis (c) origin (d) (x, y) 3. If ? is 0 zero of f(x) then ____ is one of the factors of f(x) (a) (x – ?) (b) (x – 2 ?) (c) (x + ?) (d) (2x – ?) 4. If (y – a) is factor of f(y) then ___ is a zero of f(y) (a) y (b) a (c) 2a (d) 2y 5. Which of the following is not correct for : A quadratic polynomial may have (a) no real zeros (b) two equal real zeros (c) two distinct zeros (d) three real zeros. 6. Cubic poly x = f(y) cuts y-axis at almost (a) one point (b) two points (c) three points (d) four points 7. Polynomial x 2 + 1 has ___ zeros (a) only one real (b) no real (c) only two real (d) one real and the other non-real. 8. If ?, ? are the zeros of the polynomials f (x) = x 2 + x + 1 then ? ? ? ? 1 1 ________ (a) 1 (b) –1 (c) 0 (d) none Page 3 X – Maths 9 CHAPTER 2 POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively. 2. A quadratic polynomial in x with real coefficient is of the form ax 2 + bx + c, where a, b, c are real number with a ? 0. 3. 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 i.e. x = a is a zero of polynomial p(x) if p(a) = 0. 4. A polynomial can have at most the same number of zeros as the degree of polynomial. 5. For quadratic polynomial ax 2 + bx + c (a ? 0) Sum of zeros ? ? b a Product of zeros ? . c a 6. The division algorithm states that given any polynomial p(x) and polynomial g(x), there are polynomials q(x) and r(x) such that : p(x) = g(x).q (x) + r(x), g(x) ? 0 where r(x) = 0 or degree of r(x) < degree of g(x). MULTIPLE CHOICE QUESTIONS 1. A real no. ? is a zero of the polynomial f(x) if (a) f( ?) > 0 (b) f( ?) = 0 (c) f( ?) < 0 (d) none 10 X – Maths 2. The zeros of a polynomial f(x) are the coordinates of the points where the graph of y = f(x) intersects (a) x-axis (b) y-axis (c) origin (d) (x, y) 3. If ? is 0 zero of f(x) then ____ is one of the factors of f(x) (a) (x – ?) (b) (x – 2 ?) (c) (x + ?) (d) (2x – ?) 4. If (y – a) is factor of f(y) then ___ is a zero of f(y) (a) y (b) a (c) 2a (d) 2y 5. Which of the following is not correct for : A quadratic polynomial may have (a) no real zeros (b) two equal real zeros (c) two distinct zeros (d) three real zeros. 6. Cubic poly x = f(y) cuts y-axis at almost (a) one point (b) two points (c) three points (d) four points 7. Polynomial x 2 + 1 has ___ zeros (a) only one real (b) no real (c) only two real (d) one real and the other non-real. 8. If ?, ? are the zeros of the polynomials f (x) = x 2 + x + 1 then ? ? ? ? 1 1 ________ (a) 1 (b) –1 (c) 0 (d) none X – Maths 11 9. If one of the zero of the polynomial g(x) = (k 2 + 4) x 2 + 13x + 4k is reciprocal of the other then k = ___ (a) 2 (b) – 2 (c) 1 (d) – 1 10. If 2 is a zero of both the polynomial, 3x 2 + ax – 14 and 2x – b then a – 2b = ___ (a) –2 (b) 7 (c) –8 (d) –7 11. If zeros of the polynomial ax 2 + bx + c are reciprocal of each other then (a) a = c (b) a = b (c) b = c (d) a = – c 12. The zeros of the polynomial h(x) = (x – 5) (x 2 – x–6) are (a) –2, 3, 5 (b) –2, –3, –5 (c) 2, –3, –5 (d) 2, 3, 5 13. Graph of y = ax 2 + bx + c intersects x-axis at 2 distinct points if (a) b 2 –4ac > 0 (b) b 2 – 4ac < 0 (c) b 2 –4ac = 0 (d) none SHORT ANSWER TYPE QUESTIONS 14. If ? and ? are the zeros of the polynomial 2x 2 – 7x + 3. Find the sum of the reciprocal of its zeros. 15. If ??? ? are the zeros of the polynomial p(x) = x 2 – a (x + 1) – b such that ( ? + 1) ( ? + 1) = 0 then find value of b. 16. If ??? ??are the zeros of the polynomial x 2 – (k + 6) x + 2 (2k – 1). Find k if ? ? ? ? ?? 1 . 2 17. If (x + p) is a factor of the polynomial 2x 2 + 2px + 5x + 10 find p. 18. Find a quadratic polynomial whose zeroes are ? ? ? ? ? ? 5 3 2 and 5 3 2 . Page 4 X – Maths 9 CHAPTER 2 POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively. 2. A quadratic polynomial in x with real coefficient is of the form ax 2 + bx + c, where a, b, c are real number with a ? 0. 3. 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 i.e. x = a is a zero of polynomial p(x) if p(a) = 0. 4. A polynomial can have at most the same number of zeros as the degree of polynomial. 5. For quadratic polynomial ax 2 + bx + c (a ? 0) Sum of zeros ? ? b a Product of zeros ? . c a 6. The division algorithm states that given any polynomial p(x) and polynomial g(x), there are polynomials q(x) and r(x) such that : p(x) = g(x).q (x) + r(x), g(x) ? 0 where r(x) = 0 or degree of r(x) < degree of g(x). MULTIPLE CHOICE QUESTIONS 1. A real no. ? is a zero of the polynomial f(x) if (a) f( ?) > 0 (b) f( ?) = 0 (c) f( ?) < 0 (d) none 10 X – Maths 2. The zeros of a polynomial f(x) are the coordinates of the points where the graph of y = f(x) intersects (a) x-axis (b) y-axis (c) origin (d) (x, y) 3. If ? is 0 zero of f(x) then ____ is one of the factors of f(x) (a) (x – ?) (b) (x – 2 ?) (c) (x + ?) (d) (2x – ?) 4. If (y – a) is factor of f(y) then ___ is a zero of f(y) (a) y (b) a (c) 2a (d) 2y 5. Which of the following is not correct for : A quadratic polynomial may have (a) no real zeros (b) two equal real zeros (c) two distinct zeros (d) three real zeros. 6. Cubic poly x = f(y) cuts y-axis at almost (a) one point (b) two points (c) three points (d) four points 7. Polynomial x 2 + 1 has ___ zeros (a) only one real (b) no real (c) only two real (d) one real and the other non-real. 8. If ?, ? are the zeros of the polynomials f (x) = x 2 + x + 1 then ? ? ? ? 1 1 ________ (a) 1 (b) –1 (c) 0 (d) none X – Maths 11 9. If one of the zero of the polynomial g(x) = (k 2 + 4) x 2 + 13x + 4k is reciprocal of the other then k = ___ (a) 2 (b) – 2 (c) 1 (d) – 1 10. If 2 is a zero of both the polynomial, 3x 2 + ax – 14 and 2x – b then a – 2b = ___ (a) –2 (b) 7 (c) –8 (d) –7 11. If zeros of the polynomial ax 2 + bx + c are reciprocal of each other then (a) a = c (b) a = b (c) b = c (d) a = – c 12. The zeros of the polynomial h(x) = (x – 5) (x 2 – x–6) are (a) –2, 3, 5 (b) –2, –3, –5 (c) 2, –3, –5 (d) 2, 3, 5 13. Graph of y = ax 2 + bx + c intersects x-axis at 2 distinct points if (a) b 2 –4ac > 0 (b) b 2 – 4ac < 0 (c) b 2 –4ac = 0 (d) none SHORT ANSWER TYPE QUESTIONS 14. If ? and ? are the zeros of the polynomial 2x 2 – 7x + 3. Find the sum of the reciprocal of its zeros. 15. If ??? ? are the zeros of the polynomial p(x) = x 2 – a (x + 1) – b such that ( ? + 1) ( ? + 1) = 0 then find value of b. 16. If ??? ??are the zeros of the polynomial x 2 – (k + 6) x + 2 (2k – 1). Find k if ? ? ? ? ?? 1 . 2 17. If (x + p) is a factor of the polynomial 2x 2 + 2px + 5x + 10 find p. 18. Find a quadratic polynomial whose zeroes are ? ? ? ? ? ? 5 3 2 and 5 3 2 . 12 X – Maths 19. If 1 and – 2 5 are respectively product and sum of the zeroes of a quadratic polynomial. Find the polynomial. 20. Find zeroes of ? ? 2 3 8 4 3. x x 21. If (x + k) is a factor of the polynomial x 2 –2x–15 and x 3 + a. Find k and a. 22. Form a quadratic polynomial, one of whose zero is ? ? ? 2 5 and the sum of zeros is 4. 23. If sum of the zeroes of kx 2 + 3k + 2x is equal to their product. Find k. 24. If one zero of 4x 2 – 9 – 8kx is negative of the other find k. LONG ANSWER TYPE QUESTIONS 25. Find the zeroes of 6x 2 – 3 – 7x. Verify the relationship between the zeros and coefficients. 26. If one zero of he polynomial (a 2 + a) x 2 + 13x + 6a is reciprocal of the other, find value (s) of a. 27. –5 is one of the zeroes of 2x 2 + px – 15. Quadratic polynomial p(x 2 + x) + k has both the zeros equal to each other. Then find k. 28. Find the value of k such that 3x 2 + 2kx + x – k – 5 has the sum of the zeros as half of their product. 29. If f(x) = 2x 4 – 5x 3 + x 2 + 3x – 2 is divided by g(x) the quotient is q(x) = 2x 2 – 5x + 3 and r(x) = – 2x + 1 find g(x). 30. If (x – 2) is one of the factors of x 3 – 3x 2 – 4x + 12 find the other zeros. 31. If ? and ? are the zeros of he polynomial x 2 – 5x + k such that ? – ? = 1, find the value of k. 32. If ??? ? are zeros of quadratic polynomial 2x 2 + 5x + k, find the value of k, such that ( ?? ?? ?) 2 – ?? = 24. 33. Obtain all zeros of x 4 – x 3 –7x 2 + x + 6 if 3 and 1 are zeros. 34. Find all the zeros of the polynomial 4x 4 – 20x 3 + 23x 2 + 5x – 6 if two of its zeros are 2 and 3. Page 5 X – Maths 9 CHAPTER 2 POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively. 2. A quadratic polynomial in x with real coefficient is of the form ax 2 + bx + c, where a, b, c are real number with a ? 0. 3. 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 i.e. x = a is a zero of polynomial p(x) if p(a) = 0. 4. A polynomial can have at most the same number of zeros as the degree of polynomial. 5. For quadratic polynomial ax 2 + bx + c (a ? 0) Sum of zeros ? ? b a Product of zeros ? . c a 6. The division algorithm states that given any polynomial p(x) and polynomial g(x), there are polynomials q(x) and r(x) such that : p(x) = g(x).q (x) + r(x), g(x) ? 0 where r(x) = 0 or degree of r(x) < degree of g(x). MULTIPLE CHOICE QUESTIONS 1. A real no. ? is a zero of the polynomial f(x) if (a) f( ?) > 0 (b) f( ?) = 0 (c) f( ?) < 0 (d) none 10 X – Maths 2. The zeros of a polynomial f(x) are the coordinates of the points where the graph of y = f(x) intersects (a) x-axis (b) y-axis (c) origin (d) (x, y) 3. If ? is 0 zero of f(x) then ____ is one of the factors of f(x) (a) (x – ?) (b) (x – 2 ?) (c) (x + ?) (d) (2x – ?) 4. If (y – a) is factor of f(y) then ___ is a zero of f(y) (a) y (b) a (c) 2a (d) 2y 5. Which of the following is not correct for : A quadratic polynomial may have (a) no real zeros (b) two equal real zeros (c) two distinct zeros (d) three real zeros. 6. Cubic poly x = f(y) cuts y-axis at almost (a) one point (b) two points (c) three points (d) four points 7. Polynomial x 2 + 1 has ___ zeros (a) only one real (b) no real (c) only two real (d) one real and the other non-real. 8. If ?, ? are the zeros of the polynomials f (x) = x 2 + x + 1 then ? ? ? ? 1 1 ________ (a) 1 (b) –1 (c) 0 (d) none X – Maths 11 9. If one of the zero of the polynomial g(x) = (k 2 + 4) x 2 + 13x + 4k is reciprocal of the other then k = ___ (a) 2 (b) – 2 (c) 1 (d) – 1 10. If 2 is a zero of both the polynomial, 3x 2 + ax – 14 and 2x – b then a – 2b = ___ (a) –2 (b) 7 (c) –8 (d) –7 11. If zeros of the polynomial ax 2 + bx + c are reciprocal of each other then (a) a = c (b) a = b (c) b = c (d) a = – c 12. The zeros of the polynomial h(x) = (x – 5) (x 2 – x–6) are (a) –2, 3, 5 (b) –2, –3, –5 (c) 2, –3, –5 (d) 2, 3, 5 13. Graph of y = ax 2 + bx + c intersects x-axis at 2 distinct points if (a) b 2 –4ac > 0 (b) b 2 – 4ac < 0 (c) b 2 –4ac = 0 (d) none SHORT ANSWER TYPE QUESTIONS 14. If ? and ? are the zeros of the polynomial 2x 2 – 7x + 3. Find the sum of the reciprocal of its zeros. 15. If ??? ? are the zeros of the polynomial p(x) = x 2 – a (x + 1) – b such that ( ? + 1) ( ? + 1) = 0 then find value of b. 16. If ??? ??are the zeros of the polynomial x 2 – (k + 6) x + 2 (2k – 1). Find k if ? ? ? ? ?? 1 . 2 17. If (x + p) is a factor of the polynomial 2x 2 + 2px + 5x + 10 find p. 18. Find a quadratic polynomial whose zeroes are ? ? ? ? ? ? 5 3 2 and 5 3 2 . 12 X – Maths 19. If 1 and – 2 5 are respectively product and sum of the zeroes of a quadratic polynomial. Find the polynomial. 20. Find zeroes of ? ? 2 3 8 4 3. x x 21. If (x + k) is a factor of the polynomial x 2 –2x–15 and x 3 + a. Find k and a. 22. Form a quadratic polynomial, one of whose zero is ? ? ? 2 5 and the sum of zeros is 4. 23. If sum of the zeroes of kx 2 + 3k + 2x is equal to their product. Find k. 24. If one zero of 4x 2 – 9 – 8kx is negative of the other find k. LONG ANSWER TYPE QUESTIONS 25. Find the zeroes of 6x 2 – 3 – 7x. Verify the relationship between the zeros and coefficients. 26. If one zero of he polynomial (a 2 + a) x 2 + 13x + 6a is reciprocal of the other, find value (s) of a. 27. –5 is one of the zeroes of 2x 2 + px – 15. Quadratic polynomial p(x 2 + x) + k has both the zeros equal to each other. Then find k. 28. Find the value of k such that 3x 2 + 2kx + x – k – 5 has the sum of the zeros as half of their product. 29. If f(x) = 2x 4 – 5x 3 + x 2 + 3x – 2 is divided by g(x) the quotient is q(x) = 2x 2 – 5x + 3 and r(x) = – 2x + 1 find g(x). 30. If (x – 2) is one of the factors of x 3 – 3x 2 – 4x + 12 find the other zeros. 31. If ? and ? are the zeros of he polynomial x 2 – 5x + k such that ? – ? = 1, find the value of k. 32. If ??? ? are zeros of quadratic polynomial 2x 2 + 5x + k, find the value of k, such that ( ?? ?? ?) 2 – ?? = 24. 33. Obtain all zeros of x 4 – x 3 –7x 2 + x + 6 if 3 and 1 are zeros. 34. Find all the zeros of the polynomial 4x 4 – 20x 3 + 23x 2 + 5x – 6 if two of its zeros are 2 and 3. X – Maths 13 35. If ? ? ? ? ? ? 2 3 and 2 3 are two zeroes of x 4 – 4x 3 – 8x 2 + 36x – 9 find the other two zeroes. 36. What must be subtracted from 8x 4 + 14x 3 – 4x 2 + 7x – 8 so that the resulting polynomial is exactly divisible by 4x 2 + 3x – 2. 37. When we add p(x) to 4x 4 + 2x 3 – 2x 2 + x – 1 the resulting polynomial is divisible by x 2 + 2x – 3 find p(x). 38. Find a and f if (x 4 + x 3 + 8x 2 + ax + f) is a multiple of (x 2 + 1). 39. If the polynomial 6x 4 + 8x 3 + 17x 2 + 21x + 7 is divided by 3x 2 + 1 + 4x then r(x) = (ax + b) find a and b. 40. Obtain all the zeroes of 2x 4 – 2x 3 – 7x 2 + 3x + 6 if ? ? ? ? ? ? ? ? ? 3 2 x are two factors of this polynomial. 41. Find all the zeroes of x 4 – 3x 3 – x 2 + 9x – 6 if – 3 and 3 are two of its zeros. 42. If (x 3 – 3x + 1) is one of the factors of the polynomial x 5 – 4x 3 + x 2 + 3x + 1, find the other two factors. 43. What does the graph of the polynomial ax 2 + bx + c represents. What type of graph will it represent (i) for a > 0, (ii) for a < 0. What happens if a = 0. ANSWERS 1. b 2. a 3. a 4. b 5. a 6. c 7. b 8. b 9. a 10. d 11. a 12. aRead More

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