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 t