11X1 T15 03 polynomial division (2011)

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  • 1. Polynomial Division

2. Polynomial DivisionP x A x Q x R x where; 3. Polynomial DivisionP x A x Q x R x where; A(x) is the divisor 4. Polynomial DivisionP x A x Q x R x where; A(x) is the divisorQ(x) is the quotient 5. Polynomial DivisionP x A x Q x R x where; A(x) is the divisorQ(x) is the quotient R(x) is the remainder 6. Polynomial DivisionP x A x Q x R x where; A(x) is the divisorQ(x) is the quotient R(x) is the remainderNote:degree R(x) < degree A(x) 7. Polynomial DivisionP x A x Q x R x where; A(x) is the divisorQ(x) is the quotient R(x) is the remainderNote:degree R(x) < degree A(x)Q(x) and R(x) are unique 8. Polynomial Division P x A x Q x R x where;A(x) is the divisorQ(x) is the quotientR(x) is the remainder Note: degree R(x) < degree A(x) Q(x) and R(x) are unique 1998 Extension 1 HSC Q2a)Find the quotient, Q(x), and the remainder, R(x), when the polynomialP x x 4 x 2 1 is divided by x 2 1 9. Polynomial Division P x A x Q x R x where; A(x) is the divisor Q(x) is the quotientR(x) is the remainderNote:degree R(x) < degree A(x)Q(x) and R(x) are unique 1998 Extension 1 HSC Q2a)Find the quotient, Q(x), and the remainder, R(x), when the polynomialP x x 4 x 2 1 is divided by x 2 1x 2 1 x 4 0 x3 x 2 0 x 1 10. Polynomial Division P x A x Q x R x where;A(x) is the divisorQ(x) is the quotientR(x) is the remainder Note: degree R(x) < degree A(x) Q(x) and R(x) are unique 1998 Extension 1 HSC Q2a)Find the quotient, Q(x), and the remainder, R(x), when the polynomialP x x 4 x 2 1 is divided by x 2 1x2x 2 1 x 4 0 x3 x 2 0 x 1 11. Polynomial Division P x A x Q x R x where;A(x) is the divisorQ(x) is the quotientR(x) is the remainder Note: degree R(x) < degree A(x) Q(x) and R(x) are unique 1998 Extension 1 HSC Q2a)Find the quotient, Q(x), and the remainder, R(x), when the polynomialP x x 4 x 2 1 is divided by x 2 1x2x 2 1 x 4 0 x3 x 2 0 x 1x 4 0 x3 x 2 12. Polynomial Division P x A x Q x R x where;A(x) is the divisorQ(x) is the quotientR(x) is the remainder Note: degree R(x) < degree A(x) Q(x) and R(x) are unique 1998 Extension 1 HSC Q2a)Find the quotient, Q(x), and the remainder, R(x), when the polynomialP x x 4 x 2 1 is divided by x 2 1x2x 2 1 x 4 0 x3 x 2 0 x 1x 4 0 x3 x 2 2x 2 13. Polynomial Division P x A x Q x R x where;A(x) is the divisorQ(x) is the quotientR(x) is the remainder Note: degree R(x) < degree A(x) Q(x) and R(x) are unique 1998 Extension 1 HSC Q2a)Find the quotient, Q(x), and the remainder, R(x), when the polynomialP x x 4 x 2 1 is divided by x 2 1x2x 2 1 x 4 0 x3 x 2 0 x 1x 4 0 x3 x 2 2x 2 0 x 1 14. Polynomial Division P x A x Q x R x where;A(x) is the divisorQ(x) is the quotientR(x) is the remainder Note: degree R(x) < degree A(x) Q(x) and R(x) are unique 1998 Extension 1 HSC Q2a)Find the quotient, Q(x), and the remainder, R(x), when the polynomialP x x 4 x 2 1 is divided by x 2 1x22x 2 1 x 4 0 x3 x 2 0 x 1x 4 0 x3 x 2 2x 2 0 x 1 15. Polynomial Division P x A x Q x R x where;A(x) is the divisorQ(x) is the quotientR(x) is the remainder Note: degree R(x) < degree A(x) Q(x) and R(x) are unique 1998 Extension 1 HSC Q2a)Find the quotient, Q(x), and the remainder, R(x), when the polynomialP x x 4 x 2 1 is divided by x 2 1x22x 2 1 x 4 0 x3 x 2 0 x 1x 4 0 x3 x 2 2x 2 0 x 1 2x2 0x 2 16. Polynomial Division P x A x Q x R x where;A(x) is the divisorQ(x) is the quotientR(x) is the remainder Note: degree R(x) < degree A(x) Q(x) and R(x) are unique 1998 Extension 1 HSC Q2a)Find the quotient, Q(x), and the remainder, R(x), when the polynomialP x x 4 x 2 1 is divided by x 2 1x22x 2 1 x 4 0 x3 x 2 0 x 1x 4 0 x3 x 2 2x 2 0 x 1 2x2 0x 2 17. Polynomial Division P x A x Q x R x where;A(x) is the divisorQ(x) is the quotientR(x) is the remainder Note: degree R(x) < degree A(x) Q(x) and R(x) are unique 1998 Extension 1 HSC Q2a)Find the quotient, Q(x), and the remainder, R(x), when the polynomialP x x 4 x 2 1 is divided by x 2 1x22x 2 1 x 4 0 x3 x 2 0 x 1x 4 0 x3 x 2 2x 2 0 x 1 2x2 0x 2 Q x x 2 2 and R x 3 18. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 1 19. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 1x 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 3 20. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2x 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 3 21. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2x 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 22. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2x 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 23. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2x 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 2 x 3 24. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2 4 xx 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 2 x 3 25. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2 4 xx 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 2 x 3 4 x 3 12 x 2 4 x 26. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2 4 xx 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 2 x 3 4 x 3 12 x 2 4 x 2x2 6x 27. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2 4 xx 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 2 x 3 4 x 3 12 x 2 4 x 2x2 6x 3 28. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2 4 x 2x 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 2 x 3 4 x 3 12 x 2 4 x 2x2 6x 3 29. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2 4 x 2x 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 2 x 3 4 x 3 12 x 2 4 x 2x2 6x 3 2x2 6x 2 30. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2 4 x 2x 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 2 x 3 4 x 3 12 x 2 4 x 2x2 6x 3 2x2 6x 2 1 31. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2 4 x 2x 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 2 x 3 4 x 3 12 x 2 4 x 2x2 6x 3 2x2 6x 2 1(ii) Hence write f(x) = g(x)q(x) + r(x) where q(x) and r(x) are polynomials and r(x) has degree less than 2. 32. (i) Divide the polynomial1988 Extension 1 HSC Q4b)f x 2 x 4 10 x 3 12 x 2 2 x 3 by g x x 2 3 x 12x 2 4 x 2x 2 3 x 1 2 x 4 10 x 3 12 x 2 2 x 32 x 4 6 x3 2 x 2 4 x 3 10 x 2 2 x 3 4 x 3 12 x 2 4 x 2x2 6x 3 2x2 6x 2 1(ii) Hence write f(x) = g(x)q(x) + r(x) where q(x) and r(x) are polynomials and r(x) has degree less than 2.2 x 4 10 x 3 12 x 2 2 x 3 x 2 3 x 12 x 2 4 x 2 1 33. Further graphing of polynomials P x R x y Q x A x A x 34. Further graphing of polynomials P x R x y Q x A x A x solve A(x) = 0 to find vertical asymptotes 35. Further graphing of polynomials P x R x y Q x A x A x y = Q(x) is thesolve A(x) = 0 to find horizontal/obliquevertical asymptotes asymptote 36. Further graphing of polynomialssolve R(x) = 0 to find where(if anywhere) the curve cutsthe horizontal/obliqueasymptote P x R x y Q x A x A x y = Q(x) is thesolve A(x) = 0 to find horizontal/obliquevertical asymptotes asymptote 37. 8xExample: Sketch the graph of y x 2 , clearly indicating anyx 9 asymptotes and any points where the graph meets the axes. 38. 8xExample: Sketch the graph of y x 2 , clearly indicating anyx 9 asymptotes and any points where the graph meets the axes. vertical asymptotes at x 3 39. x=-3 x=3yx 40. 8xExample: Sketch the graph of y x 2 , clearly indicating anyx 9 asymptotes and any points where the graph meets the axes. vertical asymptotes at x 3 oblique asymptote at y x 41. x=-3 x=3y y=xx 42. 8xExample: Sketch the graph of y x 2 , clearly indicating anyx 9 asymptotes and any points where the graph meets the axes. vertical asymptotes at x 3 oblique asymptote at y x curve meets oblique asymptote at 8x = 0 x=0 43. x=-3 x=3y y=xx 44. x=-3 x=3y y=xx 45. Exercise 4C; 1c, 2bdfh, 3bd, 4ac, 6ace,7b, 8, 11, 14*