Source: 0606/12/O/N/17 - Question No. 7
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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/20 - Question No. 10
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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]
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Compilation: www.mystudycompass.com
Source: 0606/13/M/J/17 - Question No. 8
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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
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p (x) = 6x 3 + ax 2 + 12x + b , where a and b are integers.
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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
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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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Compilation: www.mystudycompass.com