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5. Polynomial Theorems
Remainder Theorem
If the polynomial P(x) is divided by (x β a), then the remainder is P(a)
Proof:
P ο¨ x ο© ο½ A( x)Q( x) ο« R( x)
let A( x) ο½ ( x ο a )
P ο¨ x ο© ο½ ( x ο a )Q( x) ο« R( x)
6. Polynomial Theorems
Remainder Theorem
If the polynomial P(x) is divided by (x β a), then the remainder is P(a)
Proof:
P ο¨ x ο© ο½ A( x)Q( x) ο« R( x)
let A( x) ο½ ( x ο a )
P ο¨ x ο© ο½ ( x ο a )Q( x) ο« R( x)
P ο¨ a ο© ο½ (a ο a )Q(a ) ο« R (a )
7. Polynomial Theorems
Remainder Theorem
If the polynomial P(x) is divided by (x β a), then the remainder is P(a)
Proof:
P ο¨ x ο© ο½ A( x)Q( x) ο« R( x)
let A( x) ο½ ( x ο a )
P ο¨ x ο© ο½ ( x ο a )Q( x) ο« R( x)
P ο¨ a ο© ο½ (a ο a )Q(a ) ο« R (a )
ο½ R(a)
8. Polynomial Theorems
Remainder Theorem
If the polynomial P(x) is divided by (x β a), then the remainder is P(a)
Proof:
P ο¨ x ο© ο½ A( x)Q( x) ο« R( x)
let A( x) ο½ ( x ο a )
P ο¨ x ο© ο½ ( x ο a )Q( x) ο« R( x)
P ο¨ a ο© ο½ (a ο a )Q(a ) ο« R (a )
ο½ R(a)
now degree R ( x) οΌ 1
ο R ( x) is a constant
9. Polynomial Theorems
Remainder Theorem
If the polynomial P(x) is divided by (x β a), then the remainder is P(a)
Proof:
P ο¨ x ο© ο½ A( x)Q( x) ο« R( x)
let A( x) ο½ ( x ο a )
P ο¨ x ο© ο½ ( x ο a )Q( x) ο« R ( x)
P ο¨ a ο© ο½ (a ο a )Q(a ) ο« R (a )
ο½ R(a)
now degree R ( x) οΌ 1
ο R ( x) is a constant
R( x) ο½ R(a)
ο½ P(a)
10. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
11. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
12. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
13. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
14. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
15. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
16. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x).
17. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x).
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
18. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x).
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
ο½0
19. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x).
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
ο½0
ο ( x ο 2) is a factor
20. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x).
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
ο½0
ο ( x ο 2) is a factor
21. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x2
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
ο½0
ο ( x ο 2) is a factor
22. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x2
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0
ο ( x ο 2) is a factor
23. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x2
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2x 2
ο ( x ο 2) is a factor
24. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x2
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2x 2 ο19 x ο« 30
ο ( x ο 2) is a factor
25. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x 2 ο«2 x
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2 x 2 ο19 x ο« 30
ο ( x ο 2) is a factor
26. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x 2 ο«2 x
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2 x 2 ο19 x ο« 30
ο ( x ο 2) is a factor 2 x2 ο 4 x
27. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x 2 ο«2 x
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2 x 2 ο19 x ο« 30
ο ( x ο 2) is a factor 2 x2 ο 4 x
ο15 x
28. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x 2 ο«2 x
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2 x 2 ο19 x ο« 30
ο ( x ο 2) is a factor 2 x2 ο 4 x
ο15 x ο«30
29. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x 2 ο«2 x ο15
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2 x 2 ο19 x ο« 30
ο ( x ο 2) is a factor 2 x2 ο 4 x
ο15 x ο«30
30. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x 2 ο«2 x ο15
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2 x 2 ο19 x ο« 30
ο ( x ο 2) is a factor 2 x2 ο 4 x
ο15 x ο«30
ο15 x ο« 30
31. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x 2 ο«2 x ο15
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2 x 2 ο19 x ο« 30
ο ( x ο 2) is a factor 2 x2 ο 4 x
ο15 x ο«30
ο15 x ο« 30
0
32. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x 2 ο«2 x ο15
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2 x 2 ο19 x ο« 30
ο ( x ο 2) is a factor 2 x2 ο 4 x
ο P ( x) ο½ ( x ο 2) ο¨ x 2 ο« 2 x ο 15 ο© ο15 x ο«30
ο15 x ο« 30
0
33. e.g. Find the remainder when P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11 is divided
by (x β 2)
P ο¨ x ο© ο½ 5 x 3 ο 17 x 2 ο x ο« 11
P ο¨ 2 ο© ο½ 5 ο¨ 2 ο© ο 17 ο¨ 2 ο© ο 2 ο« 11
3 2
ο½ ο19
ο remainder when P ( x ) is divided by ( x ο 2) is ο 19
Factor Theorem
If (x β a) is a factor of P(x) then P(a) = 0
e.g. (i) Show that (x β 2) is a factor of P ο¨ x ο© ο½ x 3 ο 19 x ο« 30 and hence
factorise P(x). x 2 ο«2x ο15
x ο 2 x 3 ο« 0 x 2 ο 19 x ο« 30
P ο¨ 2 ο© ο½ ο¨ 2 ο© ο 19 ο¨ 2 ο© ο« 30
3
x3 ο 2 x 2
ο½0 2x 2 ο19 x ο« 30
ο ( x ο 2) is a factor 2 x2 ο 4 x
ο P ( x) ο½ ( x ο 2) ο¨ x 2 ο« 2 x ο 15 ο© ο15x ο«30
ο15 x ο« 30
ο½ ( x ο 2)( x ο« 5)( x ο 3)
0
34. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
35. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ ο©
36. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ ο©
leading term ο΄ leading term
=leading term
37. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ ο©
leading term ο΄ leading term
=leading term
38. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ ο©
leading term ο΄ leading term
=leading term
39. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο©
leading term ο΄ leading term
=leading term
40. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
41. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
42. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
43. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
44. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have?
45. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
46. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
How many x do you need?
47. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
How many x do you need? ο19x
48. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
How many x do you need? ο19x
How do you get from what you have to what you need?
49. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
How many x do you need? ο19x
How do you get from what you have to what you need? ο4x
50. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
How many x do you need? ο19x
How do you get from what you have to what you need? ο4x
ο4 x ο½ ο2 ο΄ ?
51. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
How many x do you need? ο19x
How do you get from what you have to what you need? ο4x
ο4 x ο½ ο2 ο΄ ?
52. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο«2x ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
How many x do you need? ο19x
How do you get from what you have to what you need? ο4x
ο4 x ο½ ο2 ο΄ ?
ο P ( x) ο½ ( x ο 2) ο¨ x 2 ο« 2 x ο 15 ο©
ο½ ( x ο 2)( x ο« 5)( x ο 3)
53. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο«2x ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
How many x do you need? ο19x
How do you get from what you have to what you need? ο4x
ο4 x ο½ ο2 ο΄ ?
54. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο«2x ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
How many x do you need? ο19x
How do you get from what you have to what you need? ο4x
ο4 x ο½ ο2 ο΄ ?
ο P ( x) ο½ ( x ο 2) ο¨ x 2 ο« 2 x ο 15 ο©
55. OR P ο¨ x ο© ο½ x 3 ο 19 x ο« 30
ο½ ( x ο 2) ο¨ x 2 ο«2x ο15 ο©
leading term ο΄ leading term constant ο΄ constant
=leading term =constant
If you where to expand out now, how many x would you have? ο15x
How many x do you need? ο19x
How do you get from what you have to what you need? ο4x
ο4 x ο½ ο2 ο΄ ?
ο P ( x) ο½ ( x ο 2) ο¨ x 2 ο« 2 x ο 15 ο©
ο½ ( x ο 2)( x ο« 5)( x ο 3)
57. (ii) Factorise P ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 36
Constant factors must be a factor of the constant
58. (ii) Factorise P ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 36
Constant factors must be a factor of the constant
Possibilities = 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36
59. (ii) Factorise P ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ 4 ο© ο½ 4 ο¨ 4 ο© ο 16 ο¨ 4 ο© ο 9 ο¨ 4 ο© ο« 36
3 2
68. (ii) Factorise P ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ 4 ο© ο½ 4 ο¨ 4 ο© ο 16 ο¨ 4 ο© ο 9 ο¨ 4 ο© ο« 36
3 2
ο½0
69. (ii) Factorise P ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ 4 ο© ο½ 4 ο¨ 4 ο© ο 16 ο¨ 4 ο© ο 9 ο¨ 4 ο© ο« 36
3 2
ο½0
ο ( x ο 4) is a factor
70. (ii) Factorise P ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ 4 ο© ο½ 4 ο¨ 4 ο© ο 16 ο¨ 4 ο© ο 9 ο¨ 4 ο© ο« 36
3 2
ο½0 P( x) ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 36
ο ( x ο 4) is a factor ο½ ο¨ x ο 4 ο©ο¨ ο©
71. (ii) Factorise P ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ 4 ο© ο½ 4 ο¨ 4 ο© ο 16 ο¨ 4 ο© ο 9 ο¨ 4 ο© ο« 36
3 2
ο½0 P( x) ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 36
ο ( x ο 4) is a factor ο½ ο¨ x ο 4 ο©ο¨ 4x 2 ο©
72. (ii) Factorise P ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ 4 ο© ο½ 4 ο¨ 4 ο© ο 16 ο¨ 4 ο© ο 9 ο¨ 4 ο© ο« 36
3 2
ο½0 P( x) ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 36
ο ( x ο 4) is a factor ο½ ο¨ x ο 4 ο©ο¨ 4x 2 ο9 ο©
73. (ii) Factorise P ο¨ x ο© ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 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 ο¨ 4 ο© ο½ 4 ο¨ 4 ο© ο 16 ο¨ 4 ο© ο 9 ο¨ 4 ο© ο« 36
3 2
ο½0 P( x) ο½ 4 x 3 ο 16 x 2 ο 9 x ο« 36
ο ( x ο 4) is a factor ο½ ο¨ x ο 4 ο©ο¨ 4x 2 ο9 ο©
ο½ ο¨ x ο 4 ο©ο¨ 2 x ο« 3ο©ο¨ 2 x ο 3ο©
74. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x β 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 β 11. When P(x) is
divided by (x β 3) the remainder is 1.
(i) What is the value of b?
75. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x β 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 β 11. When P(x) is
divided by (x β 3) the remainder is 1.
(i) What is the value of b?
Pο¨ο 1ο© ο½ ο11
76. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x β 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 β 11. When P(x) is
divided by (x β 3) the remainder is 1.
(i) What is the value of b?
Pο¨ο 1ο© ο½ ο11
οb ο½ ο11
77. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x β 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 β 11. When P(x) is
divided by (x β 3) the remainder is 1.
(i) What is the value of b?
Pο¨ο 1ο© ο½ ο11
οb ο½ ο11
(ii) What is the remainder when P(x) is divided by (x + 1)(x β 3)?
78. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x β 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 β 11. When P(x) is
divided by (x β 3) the remainder is 1.
(i) What is the value of b?
Pο¨ο 1ο© ο½ ο11
οb ο½ ο11
(ii) What is the remainder when P(x) is divided by (x + 1)(x β 3)?
Pο¨3ο© ο½ 1
79. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x β 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 β 11. When P(x) is
divided by (x β 3) the remainder is 1.
(i) What is the value of b?
Pο¨ο 1ο© ο½ ο11
οb ο½ ο11
(ii) What is the remainder when P(x) is divided by (x + 1)(x β 3)?
Pο¨3ο© ο½ 1
4a ο« b ο½ 1
80. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x β 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 β 11. When P(x) is
divided by (x β 3) the remainder is 1.
(i) What is the value of b?
Pο¨ο 1ο© ο½ ο11
οb ο½ ο11
(ii) What is the remainder when P(x) is divided by (x + 1)(x β 3)?
Pο¨3ο© ο½ 1
4a ο« b ο½ 1
4a ο½ 12
aο½3
81. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x β 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 β 11. When P(x) is
divided by (x β 3) the remainder is 1.
(i) What is the value of b?
Pο¨ο 1ο© ο½ ο11
οb ο½ ο11
(ii) What is the remainder when P(x) is divided by (x + 1)(x β 3)?
Pο¨3ο© ο½ 1
4a ο« b ο½ 1 Pο¨ x ο© ο½ ο¨ x ο« 1ο©ο¨ x ο 3ο©Qο¨ x ο© ο« 3 x ο 8
4a ο½ 12
aο½3
82. 2004 Extension 1 HSC Q3b)
Let P(x) = (x + 1)(x β 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 β 11. When P(x) is
divided by (x β 3) the remainder is 1.
(i) What is the value of b?
Pο¨ο 1ο© ο½ ο11
οb ο½ ο11
(ii) What is the remainder when P(x) is divided by (x + 1)(x β 3)?
Pο¨3ο© ο½ 1
4a ο« b ο½ 1 Pο¨ x ο© ο½ ο¨ x ο« 1ο©ο¨ x ο 3ο©Qο¨ x ο© ο« 3 x ο 8
4a ο½ 12 ο Rο¨ x ο© ο½ 3x ο 8
aο½3
83. 2002 Extension 1 HSC Q2c)
Suppose x 3 ο 2 x 2 ο« a οΊ ο¨ x ο« 2 ο©Qο¨ x ο© ο« 3 where Q(x) is a polynomial.
Find the value of a.
84. 2002 Extension 1 HSC Q2c)
Suppose x 3 ο 2 x 2 ο« a οΊ ο¨ x ο« 2 ο©Qο¨ x ο© ο« 3 where Q(x) is a polynomial.
Find the value of a.
P ο¨ο 2 ο© ο½ 3
85. 2002 Extension 1 HSC Q2c)
Suppose x 3 ο 2 x 2 ο« a οΊ ο¨ x ο« 2 ο©Qο¨ x ο© ο« 3 where Q(x) is a polynomial.
Find the value of a.
P ο¨ο 2 ο© ο½ 3
ο¨ο 2ο©3 ο 2ο¨ο 2ο©2 ο« a ο½ 3
86. 2002 Extension 1 HSC Q2c)
Suppose x 3 ο 2 x 2 ο« a οΊ ο¨ x ο« 2 ο©Qο¨ x ο© ο« 3 where Q(x) is a polynomial.
Find the value of a.
P ο¨ο 2 ο© ο½ 3
ο¨ο 2ο©3 ο 2ο¨ο 2ο©2 ο« a ο½ 3
ο 16 ο« a ο½ 3
87. 2002 Extension 1 HSC Q2c)
Suppose x 3 ο 2 x 2 ο« a οΊ ο¨ x ο« 2 ο©Qο¨ x ο© ο« 3 where Q(x) is a polynomial.
Find the value of a.
P ο¨ο 2 ο© ο½ 3
ο¨ο 2ο©3 ο 2ο¨ο 2ο©2 ο« a ο½ 3
ο 16 ο« a ο½ 3
a ο½ 19
88. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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 β 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 β 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) = β 5 , show that R(4) = β 5
91. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
92. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
P(4) = (4 + 1)(4 β 4)Q(4) + R(4)
93. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
P(4) = (4 + 1)(4 β 4)Q(4) + R(4)
R(4) = β 5
94. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
P(4) = (4 + 1)(4 β 4)Q(4) + R(4)
R(4) = β 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 β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
P(4) = (4 + 1)(4 β 4)Q(4) + R(4)
R(4) = β 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
Rο¨4ο© ο½ ο5
96. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
P(4) = (4 + 1)(4 β 4)Q(4) + R(4)
R(4) = β 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
Rο¨4ο© ο½ ο5
4a ο« b ο½ ο5
97. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
P(4) = (4 + 1)(4 β 4)Q(4) + R(4)
R(4) = β 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
Rο¨4ο© ο½ ο5 Pο¨ο 1ο© ο½ 5
4a ο« b ο½ ο5
98. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
P(4) = (4 + 1)(4 β 4)Q(4) + R(4)
R(4) = β 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
Rο¨4ο© ο½ ο5 Pο¨ο 1ο© ο½ 5
4a ο« b ο½ ο5 οaο«b ο½5
99. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
P(4) = (4 + 1)(4 β 4)Q(4) + R(4)
R(4) = β 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
Rο¨4ο© ο½ ο5 Pο¨ο 1ο© ο½ 5
4a ο« b ο½ ο5 οaο«b ο½5
ο 5a ο½ ο10
a ο½ ο2
100. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
P(4) = (4 + 1)(4 β 4)Q(4) + R(4)
R(4) = β 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
Rο¨4ο© ο½ ο5 Pο¨ο 1ο© ο½ 5
4a ο« b ο½ ο5 οaο«b ο½5
ο 5a ο½ ο10
a ο½ ο2 οb ο½ 3
101. 1994 Extension 1 HSC Q4a)
When the polynomial P(x) is divided by (x + 1)(x β 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) = β 5 , show that R(4) = β 5
P(x) = (x + 1)(x β 4)Q(x) + R(x)
P(4) = (4 + 1)(4 β 4)Q(4) + R(4)
R(4) = β 5
(iii) Further, when P(x) is divided by (x + 1), the remainder is 5. Find R(x)
Rο¨4ο© ο½ ο5 Pο¨ο 1ο© ο½ 5
4a ο« b ο½ ο5 οaο«b ο½5
ο 5a ο½ ο10
a ο½ ο2 οb ο½ 3 Rο¨ x ο© ο½ ο2 x ο« 3
102. 2x ο1
ο¦1οΆ
use P ο§ ο·
ο¨2οΈ
Exercise 4D; 1bc, 3ac, 4ac, 7acf, 8bdf, 9b, 12ac,
13, 14, 17, 23*
