# 11 x1 t15 04 polynomial theorems (2013)

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• 1. Polynomial Theorems

2. Polynomial Theorems Remainder Theorem If the polynomial P(x) is divided by (x a), then the remainder is P(a) 3. Polynomial Theorems Remainder Theorem If 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 Theorems Remainder 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 Theorems Remainder 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 Theorems Remainder 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 Theorems Remainder 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 Theorems Remainder 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 Theorems Remainder 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 R( 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 19 32 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 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32 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 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (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 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.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 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.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 30 3 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 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.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 30 0 3 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 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.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 30 0 ( x 2) is a factor 3 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 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 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 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x2 factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 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 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x2 factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 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 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x2 factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 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 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x2 factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 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 11P 2 5 2 17 2 2 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x 2 2 x factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 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 11P 2 5 2 17 2 2 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x 2 2 x factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor 2 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 11P 2 5 2 17 2 2 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x 2 2 x factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor 2 x2 4 x 15 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 11P 2 5 2 17 2 2 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x 2 2 x factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor 2 x2 4 x 15 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 11P 2 5 2 17 2 2 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x 2 2 x 15 factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor 2 x2 4 x 15 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 11P 2 5 2 17 2 2 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x 2 2 x 15 factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor 2 x2 4 x 15 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 11P 2 5 2 17 2 2 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x 2 2 x 15 factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor 2 x2 4 x 15 x 30 15 x 30 0 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 11P 2 5 2 17 2 2 11 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x 2 2 x 15 factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 2 x 2 0 2 x 2 19 x 30 ( x 2) is a factor 2 x2 4 x 15 x 30 P ( x) ( x 2) x 2 2 x 15 15 x 30 0 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 19 remainder when P ( x ) is divided by ( x 2) is 19 32Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0 e.g. (i) Show that (x 2) is a factor of P x x 3 19 x 30 and hence x 2 2x 15 factorise P(x). x 2 x 3 0 x 2 19 x 30 3 P 2 2 19 2 30 x3 2 x 2 0 2x 2 19 x 30 ( x 2) is a factor 2 x2 4 x 15x 30 P ( x) ( x 2) x 2 2 x 15 15 x 30 ( x 2)( x 5)( x 3) 0 34. ORP x x 3 19 x 30 35. ORP x x 3 19 x 30 ( x 2) 36. ORP x x 3 19 x 30 ( x 2) leading term leading term =leading term 37. ORP x x 3 19 x 30 ( x 2) leading term leading term =leading term 38. ORP x x 3 19 x 30 ( x 2) leading term leading term =leading term 39. ORP x x 3 19 x 30 ( x 2) x 2leading term leading term =leading term 40. ORP x x 3 19 x 30 ( x 2) x 2leading term leading term =leading term constant constant =constant 41. ORP x x 3 19 x 30 ( x 2) x 2leading term leading term =leading term constant constant =constant 42. ORP x x 3 19 x 30 ( x 2) x 2leading term leading term =leading term constant constant =constant 43. ORP x x 3 19 x 30 ( x 2) x 2leading term leading term =leading term15 constant constant =constant 44. ORP x x 3 19 x 30 ( x 2) x 2leading term leading term =leading term15 constant constant =constantIf you where to expand out now, how many x would you have? 45. ORP x x 3 19 x 30 ( x 2) x 2leading term leading term =leading term15 constant constant =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 2leading term leading term =leading term15 constant constant =constantIf you where to expand out now, how many x would you have? 15x How many x do you need? 47. ORP x x 3 19 x 30 ( x 2) x 2leading term leading term =leading term15 constant constant =constantIf you where to expand out now, how many x would you have? 15x How many x do you need?19x 48. ORP x x 3 19 x 30 ( x 2) x 2leading term leading term =leading term15 constant constant =constantIf you where to expand out now, how many x would you have? 15x How 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 2leading term leading term =leading term15 constant constant =constantIf you where to expand out now, how many x would you have? 15x How 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 215 constant constant =constantleading term leading term =leading termIf you where to expand out now, how many x would you have? 15x How many x do you need?19xHow do you get from what you have to what you need? 4 x 2 ?4x 51. ORP x x 3 19 x 30 ( x 2) x 215 constant constant =constantleading term leading term =leading termIf you where to expand out now, how many x would you have? 15x How many x do you need?19xHow do you get from what you have to what you need? 4 x 2 ?4x 52. ORP x x 3 19 x 30 ( x 2) x 2 2x 15 constant constant =constantleading term leading term =leading termIf you where to expand out now, how many x would you have? 15x How many x do you need?19xHow do you get from what you have to what you need? 4 x 2 ? P ( x) ( x 2) x 2 2 x 15 ( x 2)( x 5)( x 3)4x 53. ORP x x 3 19 x 30 ( x 2) x 2 2x 15 constant constant =constantleading term leading term =leading termIf you where to expand out now, how many x would you have? 15x How many x do you need?19xHow do you get from what you have to what you need? 4 x 2 ?4x 54. ORP x x 3 19 x 30 ( x 2) x 2 2x 15 constant constant =constantleading term leading term =leading termIf you where to expand out now, how many x would you have? 15x How many x do you need?19xHow do you get from what you have to what you need? 4 x 2 ? P ( x) ( x 2) x 2 2 x 15 4x 55. ORP x x 3 19 x 30 ( x 2) x 2 2x 15 constant constant =constantleading term leading term =leading termIf you where to expand out now, how many x would you have? 15x How many x do you need?19xHow do you get from what you have to what you need? 4 x 2 ? P ( x) ( x 2) x 2 2 x 15 ( x 2)( x 5)( x 3)4x 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 36 Constant factors must be a factor of the constant 58. (ii) Factorise P x 4 x 3 16 x 2 9 x 36 Constant factors must be a factor of the constant Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36 59. (ii) Factorise P x 4 x 3 16 x 2 9 x 36 Constant factors must be a factor of the constant Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36 of course they could be negative!!! 60. (ii) Factorise...