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Polynomials that no one can solve! Supriya Pisolkar IISER Pune April 16, 2017 S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 1 / 23 What are polynomials? S. Pisolkar (IISER Pune) Polynomials that no one can

  1. Polynomials that no one can solve! Supriya Pisolkar IISER Pune April 16, 2017 S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 1 / 23

  2. What are polynomials? S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 2 / 23

  3. What are polynomials? X + 1 S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 2 / 23

  4. What are polynomials? X + 1 X 2 + 2 S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 2 / 23

  5. What are polynomials? X + 1 X 2 + 2 3 X 3 + 2 X 2 − 5 S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 2 / 23

  6. What are polynomials? X + 1 X 2 + 2 3 X 3 + 2 X 2 − 5 In general a polynomial can be expressed as a n X n + a n − 1 X n − 1 + · · · + a 1 X + a 0 S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 2 / 23

  7. What are polynomials? X + 1 X 2 + 2 3 X 3 + 2 X 2 − 5 In general a polynomial can be expressed as a n X n + a n − 1 X n − 1 + · · · + a 1 X + a 0 where a i are some numbers. S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 2 / 23

  8. What are polynomials? X + 1 X 2 + 2 3 X 3 + 2 X 2 − 5 In general a polynomial can be expressed as a n X n + a n − 1 X n − 1 + · · · + a 1 X + a 0 where a i are some numbers. S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 2 / 23

  9. Degree of a polynomial The highest integer power n appearing in a n X n + a n − 1 X n − 1 · · · + a 2 X 2 + a 1 X + a 0 is called the degree of a polynomial. S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 3 / 23

  10. Degree of a polynomial The highest integer power n appearing in a n X n + a n − 1 X n − 1 · · · + a 2 X 2 + a 1 X + a 0 is called the degree of a polynomial. Type Example Degree Linear X + 1 1 X 2 + 2 X + 1 Quadratic 2 X 3 + 2 X Cubic 3 2 X 4 + x 3 + 2 Quartic 4 X 5 + 1 Quintic 5 S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 3 / 23

  11. History of polynomials S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 4 / 23

  12. History of polynomials Egyptians and Babylonians ( ∼ 4000 years ago ) : S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 4 / 23

  13. History of polynomials Egyptians and Babylonians ( ∼ 4000 years ago ) : Problem: Given a specific area, they were unable to calculate lengths of the sides of certain shapes, and without these lengths they were unable to design floor plan for their kingdom. S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 4 / 23

  14. Solving polynomials S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 5 / 23

  15. Solving polynomials Solving a polynomial p ( X ) means finding numbers which when substituted in place of X give zero. S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 5 / 23

  16. Solving polynomials Solving a polynomial p ( X ) means finding numbers which when substituted in place of X give zero. This is also called finding roots of a polynomial p ( X ). S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 5 / 23

  17. Solving polynomials Solving a polynomial p ( X ) means finding numbers which when substituted in place of X give zero. This is also called finding roots of a polynomial p ( X ). p ( X ) = X + 1, S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 5 / 23

  18. Solving polynomials Solving a polynomial p ( X ) means finding numbers which when substituted in place of X give zero. This is also called finding roots of a polynomial p ( X ). p ( X ) = X + 1, If we put X = − 1, S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 5 / 23

  19. Solving polynomials Solving a polynomial p ( X ) means finding numbers which when substituted in place of X give zero. This is also called finding roots of a polynomial p ( X ). p ( X ) = X + 1, If we put X = − 1, p ( − 1) = ( − 1) + 1 = 0 S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 5 / 23

  20. Solving polynomials Solving a polynomial p ( X ) means finding numbers which when substituted in place of X give zero. This is also called finding roots of a polynomial p ( X ). p ( X ) = X + 1, If we put X = − 1, p ( − 1) = ( − 1) + 1 = 0 So − 1 is a root of X + 1. S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 5 / 23

  21. Solving polynomials Solving a polynomial p ( X ) means finding numbers which when substituted in place of X give zero. This is also called finding roots of a polynomial p ( X ). p ( X ) = X + 1, If we put X = − 1, p ( − 1) = ( − 1) + 1 = 0 So − 1 is a root of X + 1. Let p ( X ) = ( X − 5) 2 = ( X − 5) · ( X − 5), S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 5 / 23

  22. Solving polynomials Solving a polynomial p ( X ) means finding numbers which when substituted in place of X give zero. This is also called finding roots of a polynomial p ( X ). p ( X ) = X + 1, If we put X = − 1, p ( − 1) = ( − 1) + 1 = 0 So − 1 is a root of X + 1. Let p ( X ) = ( X − 5) 2 = ( X − 5) · ( X − 5), Substitute, X = 5 S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 5 / 23

  23. Solving polynomials Solving a polynomial p ( X ) means finding numbers which when substituted in place of X give zero. This is also called finding roots of a polynomial p ( X ). p ( X ) = X + 1, If we put X = − 1, p ( − 1) = ( − 1) + 1 = 0 So − 1 is a root of X + 1. Let p ( X ) = ( X − 5) 2 = ( X − 5) · ( X − 5), Substitute, X = 5 p (5) = (5 − 5) 2 = 0, So, 5 is a root of ( X − 5) 2 . S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 5 / 23

  24. Square root of a number S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 6 / 23

  25. Square root of a number For a positive integer a , there are two numbers √ a and −√ a such that ( √ a ) 2 = a and ( −√ a ) 2 = a S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 6 / 23

  26. Square root of a number For a positive integer a , there are two numbers √ a and −√ a such that ( √ a ) 2 = a and ( −√ a ) 2 = a So, these two numbers are roots of the polynomial X 2 = a S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 6 / 23

  27. Square root of a number For a positive integer a , there are two numbers √ a and −√ a such that ( √ a ) 2 = a and ( −√ a ) 2 = a So, these two numbers are roots of the polynomial X 2 = a They are called square roots of a . S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 6 / 23

  28. Square root of a number For a positive integer a , there are two numbers √ a and −√ a such that ( √ a ) 2 = a and ( −√ a ) 2 = a So, these two numbers are roots of the polynomial X 2 = a They are called square roots of a . √ (4) 2 = 16 so, we write 16 = 4. The sign √· is called a radical sign . S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 6 / 23

  29. Solving a general Quadratic polynomials S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 7 / 23

  30. Solving a general Quadratic polynomials Consider a simple quadratic polynomial p ( X ) = X 2 + 5 X + 6. S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 7 / 23

  31. Solving a general Quadratic polynomials Consider a simple quadratic polynomial p ( X ) = X 2 + 5 X + 6. How to find a root of this equation? S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 7 / 23

  32. Solving a general Quadratic polynomials Consider a simple quadratic polynomial p ( X ) = X 2 + 5 X + 6. How to find a root of this equation? Nearly 1400 years ago Brahmagupta, an Indian mathematician, gave the first explicit root of p ( X ). S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 7 / 23

  33. Solving a general Quadratic polynomials Consider a simple quadratic polynomial p ( X ) = X 2 + 5 X + 6. How to find a root of this equation? Nearly 1400 years ago Brahmagupta, an Indian mathematician, gave the first explicit root of p ( X ). S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 7 / 23

  34. Roots of a quadratic polynomials In general, for a quadratic equation aX 2 + bX + c , there are two roots; S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 8 / 23

  35. Roots of a quadratic polynomials In general, for a quadratic equation aX 2 + bX + c , there are two roots; Muhammad ibn Musa al-Khwarizmi (Baghdad, 1100 years ago), inspired by Brahmagupta, developed a set of formulas that worked for √ √ b 2 − 4 ac ) b 2 − 4 ac ) x = − b + ( x = − b − ( , 2 a 2 a S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 8 / 23

  36. Solving Cubic polynomials S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 9 / 23

  37. Solving Cubic polynomials Gerolamo Cardano ( Italy, 600 years ago ) first published the formula for roots of cubic polynomials Figure: Gerolamo Cardano (1501-1576) S. Pisolkar (IISER Pune) Polynomials that no one can solve! April 16, 2017 9 / 23

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