# 11x1 t13 04 polynomial theorems

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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 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) let A( x) ( x a ) 5. 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) let A( x) ( x a )P x ( x a )Q( x) R( x) 6. 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) 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 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) 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 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) 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 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) 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 dividedby (x 2) 11. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (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 dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 1132 13. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 1132 19 14. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 32 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 dividedby (x 2) P x 5 x 3 17 x 2 x 11P 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) = 0 16. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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 hencefactorise P(x). 17. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19 Factor 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 hencefactorise 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 dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19 Factor 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 hencefactorise P(x).P 2 2 19 2 30 30 19. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 113 2 19 remainder when P ( x ) is divided by ( x 2) is 19 Factor 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 hencefactorise P(x).P 2 2 19 2 30 30 ( x 2) is a factor 20. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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 30P 2 2 19 2 30 30 ( x 2) is a factor 21. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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). x2x 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 30 ( x 2) is a factor 22. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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). x2x 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3 x3 2 x 20 ( x 2) is a factor 23. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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). x2x 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3 x3 2 x 20 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 dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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). x2x 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3 x3 2 x 20 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 dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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 2 xx 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3x3 2 x 20 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 dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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 2 xx 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3x3 2 x 20 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 dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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 2 xx 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3x3 2 x 20 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x 15 x 28. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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 2 xx 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3x3 2 x 20 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x 15 x 30 29. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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 2 x 15x 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3x3 2 x 20 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x 15 x 30 30. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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 2 x 15x 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3x3 2 x 20 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x 15 x 3015 x 30 31. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 3 2 19 remainder when P ( x ) is divided by ( x 2) is 19Factor 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 2 x 15x 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3x3 2 x 20 2 x 2 19 x 30 ( x 2) is a factor2 x2 4 x 15 x 3015 x 30 0 32. e.g. Find the remainder when P x 5 x 3 17 x 2 x 11 is dividedby (x 2) P x 5 x 3 17 x 2 x 11P 2 5 2 17 2 2 11 32 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 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 2 x 15x 2 x 3 0 x 2 19 x 30P 2 2 19 2 30 3x3 2 x 20 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 3015 x 30 0 33. e.g. Find the