4A05 More about Polynomials
Advanced
MCQ
Training
Advanced MCQ Training
4A Chapter 5
More about Polynomials
Question 1
Let k be a positive integer. When 3x2k + 1 + 2kx + k is divided by x + 1, the remainder
is
A. –k – 3.
B. –k + 3.
C. 3k – 3.
D. 3k + 3.
Question 2
When f (x) = x2023 + x2022 + x2021 + … + x + 1 is divided by x + 1, the remainder is
A. –1.
B. 0.
C. 1.
D. 2024.
Question 3
Let k be a constant. When the polynomial x3 + kx2 + 5 is divided by x – 5, the
remainder is twice that of divided by x + 2. Find the value of k.
A. –8
B. –4
C. 2
D. 5
Question 4
Let p(x) be a polynomial. When p(x) is divided by x2 – x – 6, the remainder is 3x – 2.
Find the remainder when p(x) is divided by x – 3.
A. –11
B. –2
2
C.
3
D. 7
Question 5
Let p(x) = ax3 + 3x2 + bx – 1, where a and b are constants. When p(x) is divided by
x + 2, the remainder is 5. Find the remainder when p(x) is divided by x – 2.
A. 16
B. 17
C. 18
D. 19
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4A05 More about Polynomials
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Question 6
Let p(x) = 4ax3 + bx2 – 7bx + 5a, where a and b are constants. When p(x) is divided by
x – 1, the remainder is –3. Find the remainder when p(x) is divided by x + 2.
A. 3
B. 5
C. 7
D. 9
Question 7
Let f (x) = px5 + qx3 + rx – 8, where p, q and r are constants. When f (x) is divided by
2x + 1, the remainder is –9. Find the value of 2p + 8q + 32r.
A. –64
B. –32
C. 32
D. 64
Question 8
Let f (x) be a polynomial. When f (x) is divided by (x – 3)2, the remainder is 2x – 5.
Find the remainder when f (x) is divided by x – 3.
A. –5
B. 0
C. 1
D. 6
Question 9
Let p(x) be a polynomial. When p(x) is divided by 2x2 – 5x – 3, the remainder is
4x + a, where a is a constant. When p(x) is divided by x – 3, the remainder is 17. Find
the remainder when p(x) is divided by 2x + 1.
A. 3
B. 5
C. 7
D. 9
Question 10
Let p(x) be a polynomial. When p(x) is divided by x + 2, the remainder is 1. When
p(x) is divided by x – 1, the remainder is 10. Find the remainder when p(x) is divided
by x2 + x – 2.
A. –3x + 7
B. –2x + 5
C. 2x + 5
D. 3x + 7
Question 11
Let p(x) be a polynomial. When p(x) is divided by x – 1, the quotient and the
remainder are q(x) and 2 respectively. It is known that when q(x) is divided by x – 3,
the remainder is 4. Find the remainder when p(x) is divided by x – 3.
A. 4
B. 6
C. 8
D. 10
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4A05 More about Polynomials
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Question 12
Let p(x) be a polynomial of degree 3. It is known that when p(x) is divided by
x2 – x – 3, the quotient and the remainder are the same. Which of the following must
be factors of p(x)?
I.
x+1
II.
x–2
III.
x–3
A. I and II only
B. I and III only
C. II and III only
D. I, II and III
Question 13
It is given that f (x) = x3 + 4x2 – 15x + 2. Let g(x) = ax4 + bx3 – 41x2 + cx + 4, where a,
b and c are constants. When g(x) is divided by f (x), the remainder is twice the
quotient. Find the remainder when g(x) is divided by f (x).
A. 3x + 1
B. 4x + 3
C. 6x + 2
D. 8x + 6
Question 14
When a polynomial f (x) is divided by x2 + x – 12, the quotient is x + 5. When f (x) is
divided by 3x, the remainder is –51. When f (x) is divided by x + 2, the remainder is
–17. Find f (x).
A. x3 + 4x2 – 13x – 51
B. x3 + 4x2 – 22x – 69
C. x3 + 6x2 – 9x – 51
D. x3 + 6x2 – 18x – 69
Question 15
Let p(x) be a polynomial. When p(x) is divided by (x + 1)(x + 2), the remainder is
x – 6. When p(x) is divided by (x + 1)(x + 3), the remainder is 2x – 5. Find the
remainder when p(x) is divided by (x + 2)(x + 3).
A. 2x – 3
B. 2x + 3
C. 3x – 2
D. 3x + 2
Question 16
Let p(x) be a polynomial and a is a constant. When p(x) is divided by x2 – x – 2, the
remainder is 7x + 10. When p(x) is divided by x2 + x – 6, the remainder is 5x + a. Find
the remainder when p(x) is divided by x2 + 4x + 3.
A. x – 4
B. x + 4
C. 2x – 5
D. 2x + 5
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4A05 More about Polynomials
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Question 17
Let p(x) be a polynomial. When p(x) is divided by x3 + 1, the remainder is x2 – 1. Find
the remainder when p(x) is divided by x2 – x + 1.
A. x – 1
B. x – 2
C. x + 1
D. x + 2
Question 18
Let k be a constant. If f (x) = x3 – 19x + 30 is divisible by x2 + 3x + k, then k =
A. –10.
B. –4.
C. 6.
D. 15.
Question 19
Let f (x) be a polynomial and a is a constant. When f (x) is divided by x2 – 3x – 28, the
remainder is ax + 32. If f (x) is divisible by x + 4, then a =
A. –8.
B. –7.
C. 7.
D. 8.
Question 20
It is known that p(x) is a polynomial which is divisible by x + 3. Which of the
following must be a factor of p(2x + 5)?
A. x + 2
B. x + 3
C. x + 4
D. x + 5
Question 21
It is known that p(x) is a polynomial which is divisible by x – 3. Find the value of the
constant a such that p(x + a) must be divisible by x + 2.
A. 5
B. 6
C. 7
D. 8
Question 22
1
Let f (x) be a cubic polynomial. It is given that f   = 0 , f (1) = –60, f (–3) = 168
2
and f (5) = 0. Find the remainder when f (x) is divided by x – 6.
A. –858
B. –210
C. 210
D. 330
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4A05 More about Polynomials
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Question 23
Let m, h and k be integers with k > h > m > 1. If (mh – 2)(hk – 2)(mk – 2) is divisible
by mhk, find the remainder when 4mh + 4hk + 4mk – 8 is divided by mhk.
A. –8
B. 0
C. 4
D. 8
Question 24
Let x be a positive integer greater than 1 and x < 100. If (x324 – 1) is divisible by
(x – 1)2, find the maximum value of x.
A. 81
B. 82
C. 96
D. 97
Question 25 NF
( x 4 − 8 x)( x 4 − 2 x3 + 4 x 2 )
Simplify 4
.
( x − 4 x 2 )( x 4 + 4 x 2 + 16)
x−2
A.
x
x−2
B.
x+2
x
C.
x−2
x
D.
x+2
Question 26 NF
1
Simplify
.
x
+ 12
3
−1
x
x −3
A. −
x − 12
x −3
B. −
( x − 6) 2
x −1
C.
x − 12
x −1
D.
( x − 6) 2
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4A05 More about Polynomials
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Question 27 NF
Find the H.C.F. of x3 + 1 and x4 + x2 + 1.
A. 1
B. x + 1
C. x2 – x + 1
D. x2 + x + 1
Question 28 NF
Find the L.C.M. of m4 – 81 and m3 – 27.
A. m – 3
B. (m – 3)2(m + 3)4
C. (m – 3)(m + 3)(m2 + 9)(m2 + 3m + 9)
D. (m – 3)2(m + 3)(m2 + 9)(m2 + 3m + 9)
Question 29 NF
Let f (x) = x3 – 7x2 + 7x + 15 and g(x) = x3 + 2x2 + kx – 12, where k is a constant. If the
H.C.F. of f (x) and g(x) is (x + 1)(x – 3), find the L.C.M. of f (x) and g(x).
A. (x + 1)(x – 2)(x – 3)(x + 5)
B. (x + 1)(x + 2)(x – 3)(x – 5)
C. (x + 1)(x – 3)(x – 4)(x + 5)
D. (x + 1)(x – 3)(x + 4)(x – 5)
Question 30 NF
Let a and b be constants. It is given that 2x3 + 11x2 + ax + b is the L.C.M. of
x2 + 4x – 12 and another polynomial. Which of the following may be the polynomial?
I.
2x + 5
II.
(x – 2)(2x + 3)
III.
(x + 6)(2x + 3)
A. I only
B. II only
C. I and III only
D. II and III only
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