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# 11 x1 t15 03 polynomial division (2013)

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### 11 x1 t15 03 polynomial division (2013)

1. 1. Polynomial Division
2. 2. Polynomial Division P x   A x   Q x   R x  where;
3. 3. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor
4. 4. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient
5. 5. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor R(x) is the remainder Q(x) is the quotient
6. 6. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor R(x) is the remainder Note: degree R(x) < degree A(x) Q(x) is the quotient
7. 7. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor R(x) is the remainder Note: degree R(x) < degree A(x) Q(x) and R(x) are unique Q(x) is the quotient
8. 8. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient R(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 polynomial P x   x 4  x 2  1 is divided by x 2  1
9. 9. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient R(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 polynomial P x   x 4  x 2  1 is divided by x 2  1 x 2  1 x 4  0 x3  x 2  0 x  1
10. 10. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient R(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 polynomial P x   x 4  x 2  1 is divided by x 2  1 x2 x 2  1 x 4  0 x3  x 2  0 x  1
11. 11. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient R(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 polynomial P x   x 4  x 2  1 is divided by x 2  1 x2 x 2  1 x 4  0 x3  x 2  0 x  1 x 4  0 x3  x 2
12. 12. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient R(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 polynomial P x   x 4  x 2  1 is divided by x 2  1 x2 x 2  1 x 4  0 x3  x 2  0 x  1 x 4  0 x3  x 2  2x 2
13. 13. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient R(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 polynomial P x   x 4  x 2  1 is divided by x 2  1 x2 x 2  1 x 4  0 x3  x 2  0 x  1 x 4  0 x3  x 2  2x 2  0 x  1
14. 14. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient R(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 polynomial P x   x 4  x 2  1 is divided by x 2  1 x2 2 x 2  1 x 4  0 x3  x 2  0 x  1 x 4  0 x3  x 2  2x 2  0 x  1
15. 15. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient R(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 polynomial P x   x 4  x 2  1 is divided by x 2  1 x2 2 x 2  1 x 4  0 x3  x 2  0 x  1 x 4  0 x3  x 2  2x 2  0 x  1  2x2  0x  2
16. 16. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient R(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 polynomial P x   x 4  x 2  1 is divided by x 2  1 x2 2 x 2  1 x 4  0 x3  x 2  0 x  1 x 4  0 x3  x 2  2x 2  0 x  1  2x2  0x  2 3
17. 17. Polynomial Division P x   A x   Q x   R x  where; A(x) is the divisor Q(x) is the quotient R(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 polynomial P x   x 4  x 2  1 is divided by x 2  1 x2 2 x 2  1 x 4  0 x3  x 2  0 x  1 x 4  0 x3  x 2  2x 2  0 x  1  2x2  0x  2  Q x   x 2  2 and R x   3 3
18. 18. (i) Divide the polynomial 1988 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. 19. (i) Divide the polynomial 1988 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 x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3
20. 20. (i) Divide the polynomial 1988 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 2x 2 x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3
21. 21. (i) Divide the polynomial 1988 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 2x 2 x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 x 4  6 x3  2 x 2
22. 22. (i) Divide the polynomial 1988 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 2x 2 x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 x 4  6 x3  2 x 2  4 x 3  10 x 2
23. 23. (i) Divide the polynomial 1988 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 2x 2 x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 x 4  6 x3  2 x 2  4 x 3  10 x 2  2 x  3
24. 24. (i) Divide the polynomial 1988 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 2x 2  4 x x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 x 4  6 x3  2 x 2  4 x 3  10 x 2  2 x  3
25. 25. (i) Divide the polynomial 1988 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 2x 2  4 x x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 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. 26. (i) Divide the polynomial 1988 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 2x 2  4 x x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 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. 27. (i) Divide the polynomial 1988 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 2x 2  4 x x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 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. 28. (i) Divide the polynomial 1988 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 2x 2  4 x  2 x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 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. 29. (i) Divide the polynomial 1988 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 2x 2  4 x  2 x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 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. 30. (i) Divide the polynomial 1988 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 2x 2  4 x  2 x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 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. 31. (i) Divide the polynomial 1988 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 2x 2  4 x  2 x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 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. 32. (i) Divide the polynomial 1988 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 2x 2  4 x  2 x 2  3 x  1 2 x 4  10 x 3  12 x 2  2 x  3 2 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  12 x 2  4 x  2   1
33. 33. Further graphing of polynomials P x  R x  y  Q x   A x  A x 
34. 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. 35. Further graphing of polynomials P x  R x  y  Q x   A x  A x  y = Q(x) is the horizontal/oblique asymptote solve A(x) = 0 to find vertical asymptotes
36. 36. Further graphing of polynomials solve R(x) = 0 to find where (if anywhere) the curve cuts the horizontal/oblique asymptote P x  R x  y  Q x   A x  A x  y = Q(x) is the horizontal/oblique asymptote solve A(x) = 0 to find vertical asymptotes
37. 37. 8x Example: Sketch the graph of y  x  2 , clearly indicating any x 9 asymptotes and any points where the graph meets the axes.
38. 38. 8x Example: Sketch the graph of y  x  2 , clearly indicating any x 9 asymptotes and any points where the graph meets the axes. • vertical asymptotes at x  3
39. 39. x=-3 x=3 y  x     
40. 40. 8x Example: Sketch the graph of y  x  2 , clearly indicating any x 9 asymptotes and any points where the graph meets the axes. • vertical asymptotes at x  3 • oblique asymptote at y  x
41. 41. x=-3 x=3 y y=x  x     
42. 42. 8x Example: Sketch the graph of y  x  2 , clearly indicating any x 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. 43. x=-3 x=3 y y=x  x     
44. 44. x=-3 x=3 y y=x  x     
45. 45. Exercise 4C; 1c, 2bdfh, 3bd, 4ac, 6ace, 7b, 8, 11, 14*