11X1 t15 04 polynomial theorems

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  • 1. Polynomial Theorems

2. Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a) 3. Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)Proof: P x A( x)Q( x) R( x) 4. Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)Proof: P x A( x)Q( x) R( x)let A( x) ( x a ) 5. Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)Proof: P x A( x)Q( x) R( x)let A( x) ( x a ) P x ( x a )Q( x) R( x) 6. Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)Proof: P x A( x)Q( x) R( x)let A( x) ( x a ) P x ( x a )Q( x) R( x) P a (a a )Q(a ) R (a ) 7. Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)Proof: P x A( x)Q( x) R( x)let A( x) ( x a ) P x ( x a )Q( x) R( x) P a (a a )Q(a ) R (a ) R(a) 8. Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)Proof: P x A( x)Q( x) R( x)let A( x) ( x a )P x ( x a )Q( x) R( x)P a (a a )Q(a ) R (a ) R(a) now degree R ( x) 1 R ( x) is a constant 9. Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)Proof: P x A( x)Q( x) R( x)let A( x) ( x a )P x ( x a )Q( x) R ( x)P a (a a )Q(a ) R (a ) R(a) now degree R ( x) 1 R ( x) is a constantR( x) R(a) P(a) 10. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2) 11. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 12. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 11 32 13. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 11 32 19 14. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 1132 19 remainder when P ( x ) is divided by ( x 2) is 19 15. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 1132 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0 16. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence factorise P(x). 17. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19 Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence factorise P(x). P 2 2 19 2 303 18. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19 Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence factorise P(x). P 2 2 19 2 303 0 19. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19 Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence factorise P(x). P 2 2 19 2 303 0 ( x 2) is a factor 20. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303 0 ( x 2) is a factor 21. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x2 x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303 0 ( x 2) is a factor 22. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x2 x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303x3 2 x 2 0 ( x 2) is a factor 23. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x2 x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303x3 2 x 2 0 2x 2 ( x 2) is a factor 24. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x2 x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303x3 2 x 2 0 2x 2 19 x 30 ( x 2) is a factor 25. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x 2 2 x x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor 26. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x 2 2 x x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x 27. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x 2 2 x x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x15 x 28. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x 2 2 x x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x15 x 30 29. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x 2 2 x 15 x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x15 x 30 30. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x 2 2 x 15 x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x15 x 30 15 x 30 31. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor Theorem If (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x 2 2 x 15 x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x15 x 30 15 x 300 32. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 1132 19 remainder when P ( x ) is divided by ( x 2) is 19Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x). x 2 2 x 15 x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor 2 x2 4 x P ( x) ( x 2) x 2 2 x 15 15 x 30 15 x 300 33. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is divided by (x 2)P x 5 x 3 17 x 2 x 11 P 2 5 2 17 2 2 11 32 19 remainder when P ( x ) is divided by ( x 2) is 19Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hencefactorise P(x).x 2 2x 15x 2 x 3 0 x 2 19 x 30 P 2 2 19 2 303x3 2 x 2 02x 2 19 x 30 ( x 2) is a factor2 x2 4 x P ( x) ( x 2) x 2 2 x 15 15x 3015 x 30 ( x 2)( x 5)( x 3) 0 34. OR P x x 3 19 x 30 35. OR P x x 3 19 x 30 ( x 2) 36. OR P x x 3 19 x 30 ( x 2) leading term leading term=leading term 37. OR P x x 3 19 x 30 ( x 2) leading term leading term=leading term 38. OR P x x 3 19 x 30 ( x 2) leading term leading term=leading term 39. OR P x x 3 19 x 30 ( x 2) x 2leading term leading term=leading term 40. OR P x x 3 19 x 30 ( x 2) x 2leading term leading termconstant constant=leading term=constant 41. OR P x x 3 19 x 30 ( x 2) x 2leading term leading termconstant constant=leading term=constant 42. OR P x x 3 19 x 30 ( x 2) x 2leading term leading termconstant constant=leading term=constant 43. OR P x x 3 19 x 30 ( x 2) x 215 leading term leading termconstant constant=leading term=constant 44. ORP x x 3 19 x 30 ( x 2) x 215 leading term leading termconstant constant =leading term=constantIf you where to expand out now, how many x would you have? 45. ORP x x 3 19 x 30 ( x 2) x 215 leading term leading termconstant constant =leading term=constantIf you where to expand out now, how many x would you have? 15x 46. ORP x x 3 19 x 30 ( x 2) x 215 leading term leading termconstant constant =leading term=constantIf you where to expand out now, how many x would you have? 15xHow many x do you need? 47. ORP x x 3 19 x 30 ( x 2) x 2 15 leading term leading term constant constant =leading term =constantIf you where to expand out now, how many x would you have? 15xHow many x do you need? 19x 48. ORP x x 3 19 x 30 ( x 2) x 2 15 leading term leading term constant constant =leading term =constantIf you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need? 49. ORP x x 3 19 x 30 ( x 2) x 2 15 leading term leading term constant constant =leading term =constantIf you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need?4x 50. ORP x x 3 19 x 30 ( x 2) x 2 15 leading term leading term constant constant =leading term =constantIf you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need?4x4 x 2 ? 51. ORP x x 3 19 x 30 ( x 2) x 2 15 leading term leading term constant constant =leading term =constantIf you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need?4x4 x 2 ? 52. ORP x x 3 19 x 30 ( x 2) x 2 2x 15 leading term leading term constant constant =leading term =constantIf you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need?4x 4 x 2 ? P ( x) ( x 2) x 2 2 x 15 ( x 2)( x 5)( x 3) 53. ORP x x 3 19 x 30 ( x 2) x 2 2x 15 leading term leading term constant constant =leading term =constantIf you where to expand out now, how many x would you have? 15xHow many x do you need?19xHow do you get from what you have to what you need?4x 4 x 2 ? 54. ORP x x 3 19 x 30 ( x 2) x 2 2x 15 leading term leading term constant constant =leading term =constantIf you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need?4x 4 x 2 ? P ( x) ( x 2) x 2 2 x 15 55. ORP x x 3 19 x 30 ( x 2) x 2 2x 15 leading term leading term constant constant =leading term =constantIf you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need?4x 4 x 2 ? P ( x) ( x 2) x 2 2 x 15 ( x 2)( x 5)( x 3) 56. (ii) Factorise P x 4 x 3 16 x 2 9 x 36 57. (ii) Factorise P x 4 x 3 16 x 2 9 x 36Constant factors must be a factor of the constant 58. (ii) Factorise P x 4 x 3 16 x 2 9 x 36Constant factors must be a factor of the constantPossibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36 59. (ii) Factorise P x 4 x 3 16 x 2 9 x 36Constant factors must be a factor of the constantPossibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36of course they could be negative!!! 60. (ii) Factorise P x 4 x 3 16 x 2 9 x 36Constant factors must be a factor of the constantPossibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18...