1. Polynomial Division

2. Polynomial Division
P x   A x   Q x   R x 
where;

3. Polynomial Division
P x   A x   Q x   R x 
where;
A(x) is the divisor

4. Polynomial Division
P x   A x   Q x   R x 
where;
A(x) is the divisor Q(x) is the quotient

5. 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

6. 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)

7. 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

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. 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. 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. 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. 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. 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. 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. 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. 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

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

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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. 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 the solve A(x) = 0 to find
horizontal/oblique vertical asymptotes
asymptote

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 solve A(x) = 0 to find
horizontal/oblique vertical asymptotes
asymptote

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. 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. x=-3 x=3
y

x
   


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. x=-3 x=3
y
y=x

x
   


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. x=-3 x=3
y
y=x

x
   


44. x=-3 x=3
y
y=x

x
   


45. Exercise 4C; 1c, 2bdfh, 3bd, 4ac, 6ace,
7b, 8, 11, 14*