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lIGCSEAlPast Year
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lv
Unit 05
m
m Factors
A
e of Polynomials
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Unit 05 Factors of polynomials
0606 Additional Mathematics
2
Mathematical Formulae
y
m
e
d
a
c
1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
n
i
v
x=
l
−b
A
2
b − 4 ac
2a
y
n
i
v
A
l
m
m
e
e
d
d
y
a
a ()
c
m
c
(
)
(
)
e
A
A
d
n
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ca
( ) vin
lv
A
l
A
n
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y
lv
m
m
A
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d
d
a
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ca
A
A
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lv
A
Binomial Theorem
A
y
n
n
n
(a + b)n = an + 1 an–1 b + 2 an–2 b2 + … + r an–r br + … + bn,
n
n!
where n is a positive integer and r =
(n – r)!r!
2. TRIGONOMETRY
Identities
sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A
Formulae for ∆ABC
a
b
c
sin A = sin B = sin C
a2 = b2 + c2 – 2bc cos A
∆=
1
bc sin A
2
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© UCLES 2015
2! of !13
0606/12/F/M/15
Unit 05 Factors of polynomials
0606 Additional Mathematics
5
0606/21/M/J/14
4
The expression 2x 3 + ax 2 + bx + 12 has a factor x - 4 and leaves a remainder of -12 when divided by
x - 1. Find the value of each of the constants a and b.
[5]
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d
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© UCLES 2014
3! of !13
0606/21/M/J/14
[Turn over
Unit 05 Factors of polynomials
0606 Additional Mathematics
4
0606/22/M/J/14
3
3x 3 - 14x 2 - 7x + d , show that d = 10.
(i) Given that x + 1 is a factor of
[1]
y
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v
A
m
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d
a
c
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A
l
(ii) Show that 3x - 14x - 7x + 10 can be written in the form ^x + 1h^ax 2 + bx + ch, where a, b
and c are constants to be found.
[2]
3
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lv
2
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ad
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c
A
A
y
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A
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ca
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A
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lv
(iii) Hence solve the equation
A
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a
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d
a
c
y
A
l
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A
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A
3x 3 - 14x 2 - 7x + 10 = 0 .
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© UCLES 2014
em
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[2]
4! of !13
0606/22/M/J/14
Unit 05 Factors of polynomials
0606 Additional Mathematics
4
0606/13/M/J/14
3
(i) Find, in terms of p, the remainder when x 3 + px 2 + p 2 x + 21 is divided by x + 3.
[2]
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d
a
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n
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A
(ii) Hence find the set of values of p for which this remainder is negative.
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[3]
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© UCLES 2014
5! of !13
0606/13/M/J/14
Unit 05 Factors of polynomials
0606 Additional Mathematics
4
0606/21/O/N/14
2
Solve the inequality 9x 2 + 2x - 1 1 (x + 1) 2 .
[3]
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3
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Solve the following simultaneous equations.
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log2 (x + y) = 3
ca
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log2 (x + 3) = 2 + log2 y
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[5]
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© UCLES 2014
6! of !13
0606/21/O/N/14
Unit 05 Factors of polynomials
0606 Additional Mathematics
14
0606/21/O/N/14
12
3x 3 - 14x 2 + 32 .
(i) Show that x - 2 is a factor of
[1]
y
(ii) Hence factorise
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ad
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y
A
completely.
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A
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ca
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3x 3 - 14x 2 + 32
c
A
A
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a
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em
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A
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a
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y
[4]
A
A
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© UCLES 2014
7! of !13
0606/21/O/N/14
Unit 05 Factors of polynomials
0606 Additional Mathematics
3
0606/23/O/N/14
1
f ^xh = 3x 3 + 8x 2 - 33x + p
The expression
has a factor of x - 2 .
(i) Show that p = 10 and express f ^xh as a product of a linear factor and a quadratic factor.
y
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d
a
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A
(ii) Hence solve the equation f (x) = 0 .
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[4]
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[2]
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www.alvinacademy.com
© UCLES 2014
8! of !13
0606/23/O/N/14
[Turn over
Unit 05 Factors of polynomials
0606 Additional Mathematics
9
0606/12/F/M/15
7
The polynomial p (x) = ax 3 + bx 2 - 3x - 4
of - 10 when divided by x + 2 .
c
A
(ii) Given that
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p (x) = (2x - 1) (rx 2 + sx + t) ,
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and leaves a remainder
y
(i) Show that a = 10 and find the value of b.
A
2x - 1
has a factor of
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l
[4]
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find the value of each of the integers r, s and t. [2]
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(iii) Hence find the exact solutions of p (x) = 0 .
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© UCLES 2015
[3]
9! of !13
0606/12/F/M/15
[Turn over
Unit 05 Factors of polynomials
0606 Additional Mathematics
16
0606/22/M/J/15
12
15x 3 + 26x 2 - 11x - 6 = 0 .
(i) Show that x = –2 is a root of the polynomial equation
[1]
y
15x 3 + 26x 2 - 11x - 6
(ii) Find the remainder when
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in
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(iii) Find the value of p and of q such that
15x 4 + px 3 - 37x 2 + qx + 6 .
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15x 3 + 26x 2 - 11x - 6
em
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is divided by x – 3.
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[2]
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is a factor of
[4]
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effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
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To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
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Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
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© UCLES 2015
! of !13
10
0606/22/M/J/15
Unit 05 Factors of polynomials
0606 Additional Mathematics
8
0606/13/M/J/15
6
The polynomial f ^xh = ax 3 - 15x 2 + bx - 2
divided by x - 1.
m
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ad
y
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A
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and a remainder of 5 when
y
(i) Show that b = 8 and find the value of a.
A
2x - 1
has a factor of
A
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em
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d
a
c
[4]
l
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(ii) Using the values of a and b from part (i), express f ^xh in the form ^2x - 1h g ^xh, where g ^xh is a
quadratic factor to be found.
[2]
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(iii) Show that the equation f ^xh = 0 has only one real root.
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© UCLES 2015
[2]
! of !13
11
0606/13/M/J/15
Unit 05 Factors of polynomials
0606 Additional Mathematics
3
0606/22/O/N/15
1
Itisgiventhat f (x) = 4x 3 - 4x 2 - 15x + 18 .
(i) Showthat x + 2 isafactorof f (x) .
y
in
A
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d
a
c
(ii) Hencefactorise f (x) completelyandsolvetheequation f (x) = 0 .
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[4]
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©UCLES2015
! of !13
12
0606/22/O/N/15
[Turn over
Unit 05 Factors of polynomials
0606 Additional Mathematics
7
0606/23/O/N/15
5
x 3 + ax 2 + bx + c = 0
The roots of the equation
value of a and of b.
are 1, 3 and 3. Show that c = –9 and find the
[4]
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© UCLES 2015
! of !13
13
0606/23/O/N/15
[Turn over