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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 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
3
2
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
19
3
2
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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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).
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
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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).
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 divided
by (x – 2)
P x 5 x 3 17 x 2 x 11
P 2 5 2 17 2 2 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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).
P 2 2 19 2 30
0
3
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
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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).
P 2 2 19 2 30
0
( x 2) is a factor
3
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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x2
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x2
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x2
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x2
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x 2 2 x
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x 2 2 x
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 2 x 2
0
2 x 2 19 x 30
( x 2) is a factor
2 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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x 2 2 x
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 2 x 2
0
2 x 2 19 x 30
( x 2) is a factor
2 x2 4 x
15 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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x 2 2 x
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 2 x 2
0
2 x 2 19 x 30
( x 2) is a factor
2 x2 4 x
15 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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x 2 2 x 15
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 2 x 2
0
2 x 2 19 x 30
( x 2) is a factor
2 x2 4 x
15 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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x 2 2 x 15
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 2 x 2
0
2 x 2 19 x 30
( x 2) is a factor
2 x2 4 x
15 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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x 2 2 x 15
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 2 x 2
0
2 x 2 19 x 30
( x 2) is a factor
2 x2 4 x
15 x 30
15 x 30
0
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 11
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x 2 2 x 15
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 2 x 2
0
2 x 2 19 x 30
( x 2) is a factor
2 x2 4 x
15 x 30
P ( x) ( x 2) x 2 2 x 15
15 x 30
0
33. 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
19
remainder when P ( x ) is divided by ( x 2) is 19
3
2
Factor 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
x 2 2x 15
factorise P(x).
x 2 x 3 0 x 2 19 x 30
3
P 2 2 19 2 30
x3 2 x 2
0
2x 2 19 x 30
( x 2) is a factor
2 x2 4 x
15x 30
P ( x) ( x 2) x 2 2 x 15
15 x 30
( x 2)( x 5)( x 3)
0
36. OR
P x x 3 19 x 30
( x 2)
leading term leading term
=leading term
37. OR
P x x 3 19 x 30
( x 2)
leading term leading term
=leading term
38. OR
P x x 3 19 x 30
( x 2)
leading term leading term
=leading term
39. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
40. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
constant constant
=constant
41. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
constant constant
=constant
42. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
constant constant
=constant
43. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
15
constant constant
=constant
44. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
15
constant constant
=constant
If you where to expand out now, how many x would you have?
45. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
15
constant constant
=constant
If you where to expand out now, how many x would you have? 15x
46. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
15
constant constant
=constant
If you where to expand out now, how many x would you have? 15x
How many x do you need?
47. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
15
constant constant
=constant
If you where to expand out now, how many x would you have? 15x
How many x do you need?
19x
48. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
15
constant constant
=constant
If you where to expand out now, how many x would you have? 15x
How many x do you need?
19x
How do you get from what you have to what you need?
49. OR
P x x 3 19 x 30
( x 2) x 2
leading term leading term
=leading term
15
constant constant
=constant
If you where to expand out now, how many x would you have? 15x
How many x do you need?
19x
How do you get from what you have to what you need?
4x
50. OR
P x x 3 19 x 30
( x 2) x 2
15
constant constant
=constant
leading term leading term
=leading term
If you where to expand out now, how many x would you have? 15x
How many x do you need?
19x
How do you get from what you have to what you need?
4 x 2 ?
4x
51. OR
P x x 3 19 x 30
( x 2) x 2
15
constant constant
=constant
leading term leading term
=leading term
If you where to expand out now, how many x would you have? 15x
How many x do you need?
19x
How do you get from what you have to what you need?
4 x 2 ?
4x
52. OR
P x x 3 19 x 30
( x 2) x 2 2x 15
constant constant
=constant
leading term leading term
=leading term
If you where to expand out now, how many x would you have? 15x
How many x do you need?
19x
How do you get from what you have to what you need?
4 x 2 ?
P ( x) ( x 2) x 2 2 x 15
( x 2)( x 5)( x 3)
4x
53. OR
P x x 3 19 x 30
( x 2) x 2 2x 15
constant constant
=constant
leading term leading term
=leading term
If you where to expand out now, how many x would you have? 15x
How many x do you need?
19x
How do you get from what you have to what you need?
4 x 2 ?
4x
54. OR
P x x 3 19 x 30
( x 2) x 2 2x 15
constant constant
=constant
leading term leading term
=leading term
If you where to expand out now, how many x would you have? 15x
How many x do you need?
19x
How do you get from what you have to what you need?
4 x 2 ?
P ( x) ( x 2) x 2 2 x 15
4x
55. OR
P x x 3 19 x 30
( x 2) x 2 2x 15
constant constant
=constant
leading term leading term
=leading term
If you where to expand out now, how many x would you have? 15x
How many x do you need?
19x
How do you get from what you have to what you need?
4 x 2 ?
P ( x) ( x 2) x 2 2 x 15
( x 2)( x 5)( x 3)
4x
57. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
58. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
59. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
60. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
Fractional factors must be of the form
61. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
62. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
63. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
64. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
1 2 3 4 6 9 12 18 36
= , , , , , , , ,
2 2 2 2 2 2 2 2 2
65. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
1 2 3 4 6 9 12 18 36
= , , , , , , , ,
2 2 2 2 2 2 2 2 2
66. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
1 2 3 4 6 9 12 18 36
= , , , , , , , ,
2 2 2 2 2 2 2 2 2 they could be negative too
67. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
1 2 3 4 6 9 12 18 36
= , , , , , , , ,
2 2 2 2 2 2 2 2 2 they could be negative too
P 4 4 4 16 4 9 4 36
3
2
68. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
1 2 3 4 6 9 12 18 36
= , , , , , , , ,
2 2 2 2 2 2 2 2 2 they could be negative too
P 4 4 4 16 4 9 4 36
0
3
2
69. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
1 2 3 4 6 9 12 18 36
= , , , , , , , ,
2 2 2 2 2 2 2 2 2 they could be negative too
P 4 4 4 16 4 9 4 36
0
( x 4) is a factor
3
2
70. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
1 2 3 4 6 9 12 18 36
= , , , , , , , ,
2 2 2 2 2 2 2 2 2 they could be negative too
P 4 4 4 16 4 9 4 36
0
( x 4) is a factor
3
2
P( x) 4 x 3 16 x 2 9 x 36
x 4
71. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
1 2 3 4 6 9 12 18 36
= , , , , , , , ,
2 2 2 2 2 2 2 2 2 they could be negative too
P 4 4 4 16 4 9 4 36
0
( x 4) is a factor
3
2
P( x) 4 x 3 16 x 2 9 x 36
x 4 4x 2
72. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
1 2 3 4 6 9 12 18 36
= , , , , , , , ,
2 2 2 2 2 2 2 2 2 they could be negative too
P 4 4 4 16 4 9 4 36
0
( x 4) is a factor
3
2
P( x) 4 x 3 16 x 2 9 x 36
x 4 4x 2
9
73. (ii) Factorise P x 4 x 3 16 x 2 9 x 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
of course they could be negative!!!
factors of the constant
Fractional factors must be of the form
factors of the leading coefficient
1 2 3 4 6 9 12 18 36
Possibilities = , , , , , , , ,
4 4 4 4 4 4 4 4 4
1 2 3 4 6 9 12 18 36
= , , , , , , , ,
2 2 2 2 2 2 2 2 2 they could be negative too
P 4 4 4 16 4 9 4 36
0
( x 4) is a factor
3
2
P( x) 4 x 3 16 x 2 9 x 36
x 4 4x 2
9
x 4 2 x 3 2 x 3
74. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x – 3)Q(x) + a(x + 1) + b, where Q(x) is a polynomial
and a and b are real numbers.
When P(x) is divided by (x + 1) the remainder is – 11. When P(x) is
divided by (x – 3) the remainder is 1.
(i) What is the value of b?
75. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x – 3)Q(x) + a(x + 1) + b, where Q(x) is a polynomial
and a and b are real numbers.
When P(x) is divided by (x + 1) the remainder is – 11. When P(x) is
divided by (x – 3) the remainder is 1.
(i) What is the value of b?
P 1 11
76. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x – 3)Q(x) + a(x + 1) + b, where Q(x) is a polynomial
and a and b are real numbers.
When P(x) is divided by (x + 1) the remainder is – 11. When P(x) is
divided by (x – 3) the remainder is 1.
(i) What is the value of b?
P 1 11
b 11
77. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x – 3)Q(x) + a(x + 1) + b, where Q(x) is a polynomial
and a and b are real numbers.
When P(x) is divided by (x + 1) the remainder is – 11. When P(x) is
divided by (x – 3) the remainder is 1.
(i) What is the value of b?
P 1 11
b 11
(ii) What is the remainder when P(x) is divided by (x + 1)(x – 3)?
78. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x – 3)Q(x) + a(x + 1) + b, where Q(x) is a polynomial
and a and b are real numbers.
When P(x) is divided by (x + 1) the remainder is – 11. When P(x) is
divided by (x – 3) the remainder is 1.
(i) What is the value of b?
P 1 11
b 11
(ii) What is the remainder when P(x) is divided by (x + 1)(x – 3)?
P3 1
79. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x – 3)Q(x) + a(x + 1) + b, where Q(x) is a polynomial
and a and b are real numbers.
When P(x) is divided by (x + 1) the remainder is – 11. When P(x) is
divided by (x – 3) the remainder is 1.
(i) What is the value of b?
P 1 11
b 11
(ii) What is the remainder when P(x) is divided by (x + 1)(x – 3)?
P3 1
4a b 1
80. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x – 3)Q(x) + a(x + 1) + b, where Q(x) is a polynomial
and a and b are real numbers.
When P(x) is divided by (x + 1) the remainder is – 11. When P(x) is
divided by (x – 3) the remainder is 1.
(i) What is the value of b?
P 1 11
b 11
(ii) What is the remainder when P(x) is divided by (x + 1)(x – 3)?
P3 1
4a b 1
4a 12
a3
81. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x – 3)Q(x) + a(x + 1) + b, where Q(x) is a polynomial
and a and b are real numbers.
When P(x) is divided by (x + 1) the remainder is – 11. When P(x) is
divided by (x – 3) the remainder is 1.
(i) What is the value of b?
P 1 11
b 11
(ii) What is the remainder when P(x) is divided by (x + 1)(x – 3)?
P3 1
4a b 1
4a 12
a3
P x x 1 x 3Q x 3 x 8
82. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x – 3)Q(x) + a(x + 1) + b, where Q(x) is a polynomial
and a and b are real numbers.
When P(x) is divided by (x + 1) the remainder is – 11. When P(x) is
divided by (x – 3) the remainder is 1.
(i) What is the value of b?
P 1 11
b 11
(ii) What is the remainder when P(x) is divided by (x + 1)(x – 3)?
P3 1
4a b 1
4a 12
a3
P x x 1 x 3Q x 3 x 8
R x 3x 8
83. 2002 Extension 1 HSC Q2c)
Suppose x 3 2 x 2 a x 2 Q x 3 where Q(x) is a polynomial.
Find the value of a.
84. 2002 Extension 1 HSC Q2c)
Suppose x 3 2 x 2 a x 2 Q x 3 where Q(x) is a polynomial.
Find the value of a.
P 2 3
85. 2002 Extension 1 HSC Q2c)
Suppose x 3 2 x 2 a x 2 Q x 3 where Q(x) is a polynomial.
Find the value of a.
P 2 3
23 2 22 a 3
86. 2002 Extension 1 HSC Q2c)
Suppose x 3 2 x 2 a x 2 Q x 3 where Q(x) is a polynomial.
Find the value of a.
P 2 3
23 2 22 a 3
16 a 3
87. 2002 Extension 1 HSC Q2c)
Suppose x 3 2 x 2 a x 2 Q x 3 where Q(x) is a polynomial.
Find the value of a.
P 2 3
23 2 22 a 3
16 a 3
a 19
88. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
89. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
90. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
91. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
92. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
P(4) = (4 + 1)(4 – 4)Q(4) + R(4)
93. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
P(4) = (4 + 1)(4 – 4)Q(4) + R(4)
R(4) = – 5
94. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
P(4) = (4 + 1)(4 – 4)Q(4) + R(4)
R(4) = – 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
95. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
P(4) = (4 + 1)(4 – 4)Q(4) + R(4)
R(4) = – 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
R4 5
96. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
P(4) = (4 + 1)(4 – 4)Q(4) + R(4)
R(4) = – 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
R4 5
4a b 5
97. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
P(4) = (4 + 1)(4 – 4)Q(4) + R(4)
R(4) = – 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
P 1 5
R4 5
4a b 5
98. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
P(4) = (4 + 1)(4 – 4)Q(4) + R(4)
R(4) = – 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
P 1 5
R4 5
ab 5
4a b 5
99. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
P(4) = (4 + 1)(4 – 4)Q(4) + R(4)
R(4) = – 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
P 1 5
R4 5
ab 5
4a b 5
5a 10
a 2
100. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
P(4) = (4 + 1)(4 – 4)Q(4) + R(4)
R(4) = – 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
P 1 5
R4 5
ab 5
4a b 5
5a 10
a 2 b 3
101. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x – 4), the quotient is
Q(x) and the remainder is R(x).
(i) Why is the most general form of R(x) given by R(x) = ax + b?
The degree of the divisor is 2, therefore the degree of the
remainder is at most 1, i.e. a linear function.
(ii) Given that P(4) = – 5 , show that R(4) = – 5
P(x) = (x + 1)(x – 4)Q(x) + R(x)
P(4) = (4 + 1)(4 – 4)Q(4) + R(4)
R(4) = – 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
P 1 5
R4 5
ab 5
4a b 5
5a 10
R x 2 x 3
a 2 b 3