CIE IGCSE ADDITIONAL MATHEMATICS
(0606)
TOPICAL PRACTICE QUESTIONS
TOPIC 5:
FACTORS OF
POLYNOMIALS
Compiled from:
Paper 1
Variants 1, 2 and 3
2016- 2020
"You will never always be
motivated, so you must
learn to be disciplined."
Source: 0606/11/M/J/16 - Question No. 10
1
Page 1
(i) Given that f (x) = 4x 3 + kx + p is exactly divisible by x + 2 and f l (x) is exactly divisible by
2x - 1 , find the value of k and of p.
[4]
(ii) Using the values of k and p found in part (i), show that f (x) = (x + 2) (ax 2 + bx + c) ,
where a, b and c are integers to be found.
[2]
(iii) Hence show that f (x) = 0 has only one solution and state this solution.
[2]
© UCLES 2016
Compilation: www.mystudycompass.com
Source: 0606/11/M/J/17 - Question No. 2
2
3
It is given that p ^xh = x 3 + ax 2 + bx - 48 . When p ^xh is divided by x - 3 the remainder is 6.
Given that pl^1h = 0 , find the value of a and of b.
(a) Simplify
Page 2
[5]
x 8 y 10 ' 3 x 3 y - 6 , giving your answer in the form x a y b , where a and b are integers. [2]
(b) (i) Show that 4 ^t - 2h2 + 5 ^t - 2h2 can be written in the form ^t - 2h p ^qt + rh, where p, q and r
are constants to be found.
[3]
1
3
(ii) Hence solve the equation 4 ^t - 2h2 + 5 ^t - 2h2 = 0 .
1
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3
[1]
Compilation: www.mystudycompass.com
Source: 0606/11/M/J/19 - Question No. 3
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Page 3
The polynomial p (x) = (2x - 1) (x + k) - 12 , where k is a constant.
(i)
Write down the value of p (- k) .
[1]
When p(x) is divided by x + 3 the remainder is 23.
(ii)
Find the value of k.
[2]
(iii)
Using your value of k, show that the equation p (x) =- 25 has no real solutions.
[3]
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Compilation: www.mystudycompass.com
Source: 0606/11/O/N/16 - Question No. 9
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Page 4
Do not use a calculator in this question.
The polynomial p(x) is ax 3 - 4x 2 + bx + 18 . It is given that p(x) and pl (x) are both divisible
by 2x - 3 .
(i) Show that a = 4 and find the value of b.
[4]
(ii) Using the values of a and b from part (i), factorise p(x) completely.
[2]
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Compilation: www.mystudycompass.com
Source: 0606/11/O/N/16 - Question No. 9
(iii) Hence find the values of x for which p(x) = x + 2 .
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[3]
Compilation: www.mystudycompass.com
Source: 0606/11/O/N/17 - Question No. 2
5
Page 6
The polynomial p(x) is ax 3 + bx 2 - 13x + 4 , where a and b are integers. Given that 2x - 1 is a
factor of p (x) and also a factor of pl (x) ,
(i) find the value of a and of b.
[5]
Using your values of a and b,
(ii) find the remainder when p (x) is divided by x + 1.
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[2]
Compilation: www.mystudycompass.com
Source: 0606/11/O/N/18 - Question No. 2
Page 7
p (x) = 2x 3 + 5x 2 + 4x + a
6
q (x) = 4x 2 + 3ax + b
Given that p(x) has a remainder of 2 when divided by 2x + 1 and that q(x) is divisible by x + 2 ,
(i)
find the value of each of the constants a and b.
[3]
Given that r (x) = p (x) - q (x) and using your values of a and b,
(ii)
find the exact remainder when r(x) is divided by 3x - 2 .
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[3]
Compilation: www.mystudycompass.com
Source: 0606/12/M/J/19 - Question No. 2
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Page 8
Do not use a calculator in this question.
Find the coordinates of the points of intersection of the curve y = (2x + 3) 2 (x - 1) and the line
y = 3 (2x + 3) .
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[5]
Compilation: www.mystudycompass.com
Source: 0606/12/O/N/16 - Question No. 9
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Page 9
Do not use a calculator in this question.
The polynomial p(x) is ax 3 - 4x 2 + bx + 18 . It is given that p(x) and pl (x) are both divisible
by 2x - 3 .
(i) Show that a = 4 and find the value of b.
[4]
(ii) Using the values of a and b from part (i), factorise p(x) completely.
[2]
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Compilation: www.mystudycompass.com
Source: 0606/12/O/N/16 - Question No. 9
(iii) Hence find the values of x for which p(x) = x + 2 .
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[3]
Compilation: www.mystudycompass.com
Source: 0606/12/O/N/17 - Question No. 7
9
Page 11
A polynomial p ^xh is ax 3 + 8x 2 + bx + 5, where a and b are integers. It is given that 2x - 1 is a
factor of p ^xh and that a remainder of –25 is obtained when p ^xh is divided by x + 2.
(i) Find the value of a and of b.
[5]
(ii) Using your values of a and b, find the exact solutions of p ^xh = 5.
[2]
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Compilation: www.mystudycompass.com
Source: 0606/12/O/N/18 - Question No. 9
10
Page 12
The polynomial p (x) = ax 3 + bx 2 + cx - 9 is divisible by x + 3. It is given that pl (0) = 36 and pll (0) = 86 .
(i)
Find the value of each of the constants a, b and c.
[6]
(ii)
Using your values of a, b and c, find the remainder when p (x) is divided by 2x - 1.
[2]
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Compilation: www.mystudycompass.com
Source: 0606/12/O/N/20 - Question No. 10
Page 13
11 The polynomial p (x) = 6x 3 + ax 2 + bx + 2 , where a and b are integers, has a factor of x - 2 .
(a) Given that p (1) =- 2p (0) , find the value of a and of b.
[4]
(b) Using your values of a and b,
(i) find the remainder when p (x) is divided by 2x - 1,
[2]
(ii) factorise p (x) .
[2]
© UCLES 2020
Compilation: www.mystudycompass.com
Source: 0606/13/M/J/16 - Question No. 10
12
Page 14
(i) Given that f (x) = 4x 3 + kx + p is exactly divisible by x + 2 and f l (x) is exactly divisible by
2x - 1 , find the value of k and of p.
[4]
(ii) Using the values of k and p found in part (i), show that f (x) = (x + 2) (ax 2 + bx + c) ,
where a, b and c are integers to be found.
[2]
(iii) Hence show that f (x) = 0 has only one solution and state this solution.
[2]
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Compilation: www.mystudycompass.com
Source: 0606/13/M/J/17 - Question No. 8
Page 15
13 It is given that p (x) = 2x 3 + ax 2 + 4x + b , where a and b are constants. It is given also that 2x + 1 is a
factor of p (x) and that when p (x) is divided by x - 1 there is a remainder of - 12 .
(i) Find the value of a and of b.
[5]
(ii) Using your values of a and b, write p (x) in the form (2x + 1) q (x) , where q (x) is a quadratic
expression.
[2]
(iii) Hence find the exact solutions of the equation p (x) = 0 .
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[2]
Compilation: www.mystudycompass.com
Source: 0606/13/M/J/20 - Question No. 5
Page 16
p (x) = 6x 3 + ax 2 + 12x + b , where a and b are integers.
14
p (x) has a remainder of 11 when divided by x - 3 and a remainder of - 21 when divided by x + 1.
(a) Given that p (x) = (x - 2) Q (x) , find Q (x) , a quadratic factor with numerical coefficients.
[6]
(b) Hence solve p (x) = 0 .
[2]
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Compilation: www.mystudycompass.com
Source: 0606/13/O/N/18 - Question No. 11
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The polynomial p (x) = ax 3 + 17x 2 + bx - 8 is divisible by 2x - 1 and has a remainder of - 35 when
divided by x + 3.
(i)
By finding the value of each of the constants a and b, verify that a = b .
[4]
Using your values of a and b,
(ii)
find p (x) in the form (2x - 1) q (x) , where q (x) is a quadratic expression,
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[2]
Compilation: www.mystudycompass.com
Source: 0606/13/O/N/18 - Question No. 11
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(iii)
factorise p (x) completely,
[1]
(iv)
solve a sin 3 i + 17 sin 2 i + b sin i - 8 = 0 for 0° 1 i 1 180° .
[3]
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Compilation: www.mystudycompass.com
Source: 0606/13/O/N/20 - Question No. 7
Page 19
16 The polynomial p (x) = ax 3 + bx 2 - 19x + 4 , where a and b are constants, has a factor x + 4 and is
such that 2p (1) = 5p (0) .
(a) Show that p (x) = (x + 4) (Ax 2 + Bx + C) , where A, B and C are integers to be found.
[6]
(b) Hence factorise p (x) .
[1]
(c) Find the remainder when pl (x) is divided by x.
[1]
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