AS PURE MATHEMATCICS WORKED SOLUTIONS: THE FACTOR THEOREM 1 Exercise 5A Question 1: a. 2 3 2 3 2 2 2 6 22 3 3 4 1 3 6 4 6 18 22 1 22 66 67 x x x x x x x x x x x x x x − + + − + − + − + − − − + − 3 2 2 67 3 4 1 6 22 3 x x x x x x ∴ − + − = − + − + The rest can be done similarly. b. 2 2 2 x x + + c. 3 2 5 32 2 2 1 x x x + + − − d. 3 2 675 9 42 169 4 x x x x − + − + + e. 4 3 2 23 3 5 11 2 x x x x x + + + + + − f. 2 5 23 165 559 64 4 16 64 4 x x x + + + + g. 2 8 4 5 2 x x x − + − + + h. 5 4 3 2 2 13 110 523 2696 11779 58166 729 3 9 27 81 243 729 3 5 x x x x x x + + + + + + − i. 4 3 2 231 2 2 9 26 79 3 x x x x x − − − − − − − j. 5 4 3 2 621174 11 95 852 7669 69020 9 x x x x x x + + + + + + − k. 2 3 5 3 58 2 4 8 1 2 x x x − − + + − l. 2 1 13 55 27 64 4 16 64 3 4 x x x − + + + −
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Solutions - Factor TheoremAS PURE MATHEMATCICS WORKED SOLUTIONS: THE FACTOR THEOREM 1 Exercise 5A Question 1: a. 2 32 32 2 2 6 22 33 41 3 6 4 6 18 22 1 22 66 67 xx xx x x xx xx xx
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AS PURE MATHEMATCICS WORKED SOLUTIONS: THE FACTOR THEOREM 1
Exercise 5A
Question 1:
a.
2
3 2
3 2
2
2
6 223 3 4 1
3 6 4 6 18 22 1 22 66 67
x xx x x x
x xx xx x
xx
− ++ − + −
+− +− −
−+−
3 2 2 673 4 1 6 223
x x x x xx
∴ − + − = − + −+
The rest can be done similarly.
b. 22 2x x+ +
c. 3 2 5 3 22 2 1
x xx
+ + −−
d. 3 2 6759 42 1694
x x xx
− + − ++
e. 4 3 2 233 5 112
x x x xx
+ + + + +−
f. 25 23 165 559 644 16 64 4x x
x+ + +
+
g. 2 84 52
x xx
− + − ++
h. 5 4 3 22 13 110 523 2696 11779 58166 7293 9 27 81 243 729 3 5x x x x x
x+ + + + + +
−
i. 4 3 2 2312 2 9 26 793
x x x xx
− − − − − −−
j. 5 4 3 2 62117411 95 852 7669 690209
x x x x xx
+ + + + + +−
k. 23 5 3 5 82 4 8 1 2x x
x− − + +
−
l. 21 13 55 27 644 16 64 3 4x x
x− + + +
−
AS PURE MATHEMATCICS WORKED SOLUTIONS: THE FACTOR THEOREM 2
Question 2:
2
3 2
3 2
2
2
5 11 4 6 2
5 6 5 5 2 1 1
x xx x x x
x xx xx x
xx
+ −− + − +
−−−− +− +
3 224 6 2 15 1
1 1x x x x x
x x+ − +∴ = + − +
− −
AS PURE MATHEMATCICS WORKED SOLUTIONS: THE FACTOR THEOREM 3
Question 3:
2
3 2
3 2
2
2
3 6 112 3 0 1
3 6 6 6 12 11 1 11 22 23
x xx x x x
x xx xx x
xx
− ++ + − −
+− −− −
−+−
( ) ( )3 2 231 2 3 6 112
x x x x xx
∴ − − ÷ + = − + −+
AS PURE MATHEMATCICS WORKED SOLUTIONS: THE FACTOR THEOREM 4
Question 4:
a.
2
3 2
3 2
2
2
8 203 5 4 1
3 8 4 8 24 20 1 22 60 61
x xx x x x
x xx xx x
xx
+ +− + − +
−−−
+−
So the remainder is 61 .
b. The remainder is 2− .
c. The remainder is 116
−
d. The remainder is 18 .
AS PURE MATHEMATCICS WORKED SOLUTIONS: THE FACTOR THEOREM 5
Exercise 5B
Question 1:
a. Let 3 2( ) 3 2 5 1f x x x x= + − + . Then
( ) ( ) ( )3 2( 2) 3 2 2 2 5 2 15
f − = − + − − − += −
So by the remainder theorem, the remainder is 5− .
b. Let 4 3( ) 2 2 5 1f x x x x= − − + + .Then
( ) ( ) ( )4 3(3) 2 3 2 3 5 3 1200
f = − − + += −
So by the remainder theorem, the remainder is 200−
c. Let 5 4 3( ) 8 4 6 4 4f x x x x x= + + − + .Then