Chapter 2: Polynomials MCQs with Solutions Class 10 Notes | EduRev

Class 10 : Chapter 2: Polynomials MCQs with Solutions Class 10 Notes | EduRev

 Page 1


X – Maths 9
CHAPTER 2
POLYNOMIALS
KEY POINTS
1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic
polynomials respectively.
2. A quadratic polynomial in x with real coefficient is of the form ax
2
 + bx + c,
where a, b, c are real number with a ? 0.
3. The zeroes of a polynomial p(x) are precisely the x–coordinates of the
points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero
of polynomial p(x) if p(a) = 0.
4. A polynomial can have at most the same number of zeros as the degree
of polynomial.
5. For quadratic polynomial ax
2
 + bx + c (a ? 0)
Sum of zeros ? ?
b
a
Product of zeros ? .
c
a
6. The division algorithm states that given any polynomial p(x) and polynomial
g(x), there are polynomials q(x) and r(x) such that :
p(x) = g(x).q (x) + r(x), g(x) ? 0
where r(x) = 0 or degree of r(x) < degree of g(x).
MULTIPLE CHOICE QUESTIONS
1. A real no. ? is a zero of the polynomial f(x) if
(a) f( ?) > 0 (b) f( ?) = 0
(c) f( ?) < 0 (d) none
Page 2


X – Maths 9
CHAPTER 2
POLYNOMIALS
KEY POINTS
1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic
polynomials respectively.
2. A quadratic polynomial in x with real coefficient is of the form ax
2
 + bx + c,
where a, b, c are real number with a ? 0.
3. The zeroes of a polynomial p(x) are precisely the x–coordinates of the
points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero
of polynomial p(x) if p(a) = 0.
4. A polynomial can have at most the same number of zeros as the degree
of polynomial.
5. For quadratic polynomial ax
2
 + bx + c (a ? 0)
Sum of zeros ? ?
b
a
Product of zeros ? .
c
a
6. The division algorithm states that given any polynomial p(x) and polynomial
g(x), there are polynomials q(x) and r(x) such that :
p(x) = g(x).q (x) + r(x), g(x) ? 0
where r(x) = 0 or degree of r(x) < degree of g(x).
MULTIPLE CHOICE QUESTIONS
1. A real no. ? is a zero of the polynomial f(x) if
(a) f( ?) > 0 (b) f( ?) = 0
(c) f( ?) < 0 (d) none
10 X – Maths
2. The zeros of a polynomial f(x) are the coordinates of the points where the
graph of y = f(x) intersects
(a) x-axis (b) y-axis
(c) origin (d) (x, y)
3. If ? is 0 zero of f(x) then ____ is one of the factors of f(x)
(a) (x – ?) (b) (x – 2 ?)
(c) (x + ?) (d) (2x – ?)
4. If (y – a) is factor of f(y) then ___ is a zero of f(y)
(a) y (b) a
(c) 2a (d) 2y
5. Which of the following is not correct for : A quadratic polynomial may
have
(a) no real zeros (b) two equal real
zeros
(c) two distinct zeros (d) three real zeros.
6. Cubic poly x = f(y) cuts y-axis at almost
(a) one point (b) two points
(c) three points (d) four points
7. Polynomial x
2
 + 1 has ___ zeros
(a) only one real (b) no real
(c) only two real (d) one real and the
other non-real.
8. If ?, ? are the zeros of the polynomials f (x) = x
2
 + x + 1 then
? ?
? ?
1 1
________
(a) 1 (b) –1
(c) 0 (d) none
Page 3


X – Maths 9
CHAPTER 2
POLYNOMIALS
KEY POINTS
1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic
polynomials respectively.
2. A quadratic polynomial in x with real coefficient is of the form ax
2
 + bx + c,
where a, b, c are real number with a ? 0.
3. The zeroes of a polynomial p(x) are precisely the x–coordinates of the
points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero
of polynomial p(x) if p(a) = 0.
4. A polynomial can have at most the same number of zeros as the degree
of polynomial.
5. For quadratic polynomial ax
2
 + bx + c (a ? 0)
Sum of zeros ? ?
b
a
Product of zeros ? .
c
a
6. The division algorithm states that given any polynomial p(x) and polynomial
g(x), there are polynomials q(x) and r(x) such that :
p(x) = g(x).q (x) + r(x), g(x) ? 0
where r(x) = 0 or degree of r(x) < degree of g(x).
MULTIPLE CHOICE QUESTIONS
1. A real no. ? is a zero of the polynomial f(x) if
(a) f( ?) > 0 (b) f( ?) = 0
(c) f( ?) < 0 (d) none
10 X – Maths
2. The zeros of a polynomial f(x) are the coordinates of the points where the
graph of y = f(x) intersects
(a) x-axis (b) y-axis
(c) origin (d) (x, y)
3. If ? is 0 zero of f(x) then ____ is one of the factors of f(x)
(a) (x – ?) (b) (x – 2 ?)
(c) (x + ?) (d) (2x – ?)
4. If (y – a) is factor of f(y) then ___ is a zero of f(y)
(a) y (b) a
(c) 2a (d) 2y
5. Which of the following is not correct for : A quadratic polynomial may
have
(a) no real zeros (b) two equal real
zeros
(c) two distinct zeros (d) three real zeros.
6. Cubic poly x = f(y) cuts y-axis at almost
(a) one point (b) two points
(c) three points (d) four points
7. Polynomial x
2
 + 1 has ___ zeros
(a) only one real (b) no real
(c) only two real (d) one real and the
other non-real.
8. If ?, ? are the zeros of the polynomials f (x) = x
2
 + x + 1 then
? ?
? ?
1 1
________
(a) 1 (b) –1
(c) 0 (d) none
X – Maths 11
9. If one of the zero of the polynomial g(x) = (k
2
 + 4) x
2
 + 13x + 4k is
reciprocal of the other then k = ___
(a) 2 (b) – 2
(c) 1 (d) – 1
10. If 2 is a zero of both the polynomial, 3x
2
 + ax – 14 and 2x – b then
a – 2b = ___
(a) –2 (b) 7
(c) –8 (d) –7
11. If zeros of the polynomial ax
2
 + bx + c are reciprocal of each other then
(a) a = c (b) a = b
(c) b = c (d) a = – c
12. The zeros of the polynomial h(x) = (x – 5) (x
2
 – x–6) are
(a) –2, 3, 5 (b) –2, –3, –5
(c) 2, –3, –5 (d) 2, 3, 5
13. Graph of y = ax
2
 + bx + c intersects x-axis at 2 distinct points if
(a) b
2
 –4ac > 0 (b) b
2
 – 4ac < 0
(c) b
2
 –4ac = 0 (d) none
SHORT ANSWER TYPE QUESTIONS
14. If ? and ? are the zeros of the polynomial 2x
2
 – 7x + 3. Find the sum of
the reciprocal of its zeros.
15. If ??? ? are the zeros of the polynomial p(x) = x
2
 – a (x + 1) – b such that
( ? + 1) ( ? + 1) = 0 then find value of b.
16. If ??? ??are the zeros of the polynomial x
2
 – (k + 6) x + 2 (2k – 1). Find
k if ? ? ? ? ??
1
.
2
17. If (x + p) is a factor of the polynomial 2x
2
 + 2px + 5x + 10 find p.
18. Find a quadratic polynomial whose zeroes are 
? ? ? ?
? ? 5 3 2 and 5 3 2 .
Page 4


X – Maths 9
CHAPTER 2
POLYNOMIALS
KEY POINTS
1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic
polynomials respectively.
2. A quadratic polynomial in x with real coefficient is of the form ax
2
 + bx + c,
where a, b, c are real number with a ? 0.
3. The zeroes of a polynomial p(x) are precisely the x–coordinates of the
points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero
of polynomial p(x) if p(a) = 0.
4. A polynomial can have at most the same number of zeros as the degree
of polynomial.
5. For quadratic polynomial ax
2
 + bx + c (a ? 0)
Sum of zeros ? ?
b
a
Product of zeros ? .
c
a
6. The division algorithm states that given any polynomial p(x) and polynomial
g(x), there are polynomials q(x) and r(x) such that :
p(x) = g(x).q (x) + r(x), g(x) ? 0
where r(x) = 0 or degree of r(x) < degree of g(x).
MULTIPLE CHOICE QUESTIONS
1. A real no. ? is a zero of the polynomial f(x) if
(a) f( ?) > 0 (b) f( ?) = 0
(c) f( ?) < 0 (d) none
10 X – Maths
2. The zeros of a polynomial f(x) are the coordinates of the points where the
graph of y = f(x) intersects
(a) x-axis (b) y-axis
(c) origin (d) (x, y)
3. If ? is 0 zero of f(x) then ____ is one of the factors of f(x)
(a) (x – ?) (b) (x – 2 ?)
(c) (x + ?) (d) (2x – ?)
4. If (y – a) is factor of f(y) then ___ is a zero of f(y)
(a) y (b) a
(c) 2a (d) 2y
5. Which of the following is not correct for : A quadratic polynomial may
have
(a) no real zeros (b) two equal real
zeros
(c) two distinct zeros (d) three real zeros.
6. Cubic poly x = f(y) cuts y-axis at almost
(a) one point (b) two points
(c) three points (d) four points
7. Polynomial x
2
 + 1 has ___ zeros
(a) only one real (b) no real
(c) only two real (d) one real and the
other non-real.
8. If ?, ? are the zeros of the polynomials f (x) = x
2
 + x + 1 then
? ?
? ?
1 1
________
(a) 1 (b) –1
(c) 0 (d) none
X – Maths 11
9. If one of the zero of the polynomial g(x) = (k
2
 + 4) x
2
 + 13x + 4k is
reciprocal of the other then k = ___
(a) 2 (b) – 2
(c) 1 (d) – 1
10. If 2 is a zero of both the polynomial, 3x
2
 + ax – 14 and 2x – b then
a – 2b = ___
(a) –2 (b) 7
(c) –8 (d) –7
11. If zeros of the polynomial ax
2
 + bx + c are reciprocal of each other then
(a) a = c (b) a = b
(c) b = c (d) a = – c
12. The zeros of the polynomial h(x) = (x – 5) (x
2
 – x–6) are
(a) –2, 3, 5 (b) –2, –3, –5
(c) 2, –3, –5 (d) 2, 3, 5
13. Graph of y = ax
2
 + bx + c intersects x-axis at 2 distinct points if
(a) b
2
 –4ac > 0 (b) b
2
 – 4ac < 0
(c) b
2
 –4ac = 0 (d) none
SHORT ANSWER TYPE QUESTIONS
14. If ? and ? are the zeros of the polynomial 2x
2
 – 7x + 3. Find the sum of
the reciprocal of its zeros.
15. If ??? ? are the zeros of the polynomial p(x) = x
2
 – a (x + 1) – b such that
( ? + 1) ( ? + 1) = 0 then find value of b.
16. If ??? ??are the zeros of the polynomial x
2
 – (k + 6) x + 2 (2k – 1). Find
k if ? ? ? ? ??
1
.
2
17. If (x + p) is a factor of the polynomial 2x
2
 + 2px + 5x + 10 find p.
18. Find a quadratic polynomial whose zeroes are 
? ? ? ?
? ? 5 3 2 and 5 3 2 .
12 X – Maths
19. If 
1
and – 2
5
 are respectively product and sum of the zeroes of a quadratic
polynomial. Find the polynomial.
20. Find zeroes of 
? ?
2
3 8 4 3. x x
21. If (x + k) is a factor of the polynomial x
2
–2x–15 and x
3
 + a. Find k and a.
22. Form a quadratic polynomial, one of whose zero is ? ?
? 2 5
 and the
sum of zeros is 4.
23. If sum of the zeroes of kx
2
 + 3k + 2x is equal to their product. Find k.
24. If one zero of 4x
2
 – 9 – 8kx is negative of the other find k.
LONG ANSWER TYPE QUESTIONS
25. Find the zeroes of 6x
2
 – 3 – 7x. Verify the relationship between the zeros
and coefficients.
26. If one zero of he polynomial (a
2
 + a) x
2
 + 13x + 6a is reciprocal of the
other, find value (s) of a.
27. –5 is one of the zeroes of 2x
2
 + px – 15. Quadratic polynomial
p(x
2
 + x) + k has both the zeros equal to each other. Then find k.
28. Find the value of k such that 3x
2
 + 2kx + x – k – 5 has the sum of the
zeros as half of their product.
29. If f(x) = 2x
4
 – 5x
3 
+ x
2
 + 3x – 2 is divided by g(x) the quotient is
q(x) = 2x
2
 – 5x + 3 and r(x) = – 2x + 1 find g(x).
30. If (x – 2) is one of the factors of x
3
 – 3x
2 
– 4x + 12 find the other zeros.
31. If ? and ? are the zeros of he polynomial x
2
 – 5x + k such that ? – ? =
1, find the value of k.
32. If ??? ? are zeros of quadratic polynomial 2x
2
 + 5x + k, find the value of
k, such that ( ?? ?? ?)
2
 – ?? = 24.
33. Obtain all zeros of x
4
 – x
3
 –7x
2
 + x + 6 if 3 and 1 are zeros.
34. Find all the zeros of the polynomial 4x
4
 – 20x
3
 + 23x
2
 + 5x – 6 if two of
its zeros are 2 and 3.
Page 5


X – Maths 9
CHAPTER 2
POLYNOMIALS
KEY POINTS
1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic
polynomials respectively.
2. A quadratic polynomial in x with real coefficient is of the form ax
2
 + bx + c,
where a, b, c are real number with a ? 0.
3. The zeroes of a polynomial p(x) are precisely the x–coordinates of the
points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero
of polynomial p(x) if p(a) = 0.
4. A polynomial can have at most the same number of zeros as the degree
of polynomial.
5. For quadratic polynomial ax
2
 + bx + c (a ? 0)
Sum of zeros ? ?
b
a
Product of zeros ? .
c
a
6. The division algorithm states that given any polynomial p(x) and polynomial
g(x), there are polynomials q(x) and r(x) such that :
p(x) = g(x).q (x) + r(x), g(x) ? 0
where r(x) = 0 or degree of r(x) < degree of g(x).
MULTIPLE CHOICE QUESTIONS
1. A real no. ? is a zero of the polynomial f(x) if
(a) f( ?) > 0 (b) f( ?) = 0
(c) f( ?) < 0 (d) none
10 X – Maths
2. The zeros of a polynomial f(x) are the coordinates of the points where the
graph of y = f(x) intersects
(a) x-axis (b) y-axis
(c) origin (d) (x, y)
3. If ? is 0 zero of f(x) then ____ is one of the factors of f(x)
(a) (x – ?) (b) (x – 2 ?)
(c) (x + ?) (d) (2x – ?)
4. If (y – a) is factor of f(y) then ___ is a zero of f(y)
(a) y (b) a
(c) 2a (d) 2y
5. Which of the following is not correct for : A quadratic polynomial may
have
(a) no real zeros (b) two equal real
zeros
(c) two distinct zeros (d) three real zeros.
6. Cubic poly x = f(y) cuts y-axis at almost
(a) one point (b) two points
(c) three points (d) four points
7. Polynomial x
2
 + 1 has ___ zeros
(a) only one real (b) no real
(c) only two real (d) one real and the
other non-real.
8. If ?, ? are the zeros of the polynomials f (x) = x
2
 + x + 1 then
? ?
? ?
1 1
________
(a) 1 (b) –1
(c) 0 (d) none
X – Maths 11
9. If one of the zero of the polynomial g(x) = (k
2
 + 4) x
2
 + 13x + 4k is
reciprocal of the other then k = ___
(a) 2 (b) – 2
(c) 1 (d) – 1
10. If 2 is a zero of both the polynomial, 3x
2
 + ax – 14 and 2x – b then
a – 2b = ___
(a) –2 (b) 7
(c) –8 (d) –7
11. If zeros of the polynomial ax
2
 + bx + c are reciprocal of each other then
(a) a = c (b) a = b
(c) b = c (d) a = – c
12. The zeros of the polynomial h(x) = (x – 5) (x
2
 – x–6) are
(a) –2, 3, 5 (b) –2, –3, –5
(c) 2, –3, –5 (d) 2, 3, 5
13. Graph of y = ax
2
 + bx + c intersects x-axis at 2 distinct points if
(a) b
2
 –4ac > 0 (b) b
2
 – 4ac < 0
(c) b
2
 –4ac = 0 (d) none
SHORT ANSWER TYPE QUESTIONS
14. If ? and ? are the zeros of the polynomial 2x
2
 – 7x + 3. Find the sum of
the reciprocal of its zeros.
15. If ??? ? are the zeros of the polynomial p(x) = x
2
 – a (x + 1) – b such that
( ? + 1) ( ? + 1) = 0 then find value of b.
16. If ??? ??are the zeros of the polynomial x
2
 – (k + 6) x + 2 (2k – 1). Find
k if ? ? ? ? ??
1
.
2
17. If (x + p) is a factor of the polynomial 2x
2
 + 2px + 5x + 10 find p.
18. Find a quadratic polynomial whose zeroes are 
? ? ? ?
? ? 5 3 2 and 5 3 2 .
12 X – Maths
19. If 
1
and – 2
5
 are respectively product and sum of the zeroes of a quadratic
polynomial. Find the polynomial.
20. Find zeroes of 
? ?
2
3 8 4 3. x x
21. If (x + k) is a factor of the polynomial x
2
–2x–15 and x
3
 + a. Find k and a.
22. Form a quadratic polynomial, one of whose zero is ? ?
? 2 5
 and the
sum of zeros is 4.
23. If sum of the zeroes of kx
2
 + 3k + 2x is equal to their product. Find k.
24. If one zero of 4x
2
 – 9 – 8kx is negative of the other find k.
LONG ANSWER TYPE QUESTIONS
25. Find the zeroes of 6x
2
 – 3 – 7x. Verify the relationship between the zeros
and coefficients.
26. If one zero of he polynomial (a
2
 + a) x
2
 + 13x + 6a is reciprocal of the
other, find value (s) of a.
27. –5 is one of the zeroes of 2x
2
 + px – 15. Quadratic polynomial
p(x
2
 + x) + k has both the zeros equal to each other. Then find k.
28. Find the value of k such that 3x
2
 + 2kx + x – k – 5 has the sum of the
zeros as half of their product.
29. If f(x) = 2x
4
 – 5x
3 
+ x
2
 + 3x – 2 is divided by g(x) the quotient is
q(x) = 2x
2
 – 5x + 3 and r(x) = – 2x + 1 find g(x).
30. If (x – 2) is one of the factors of x
3
 – 3x
2 
– 4x + 12 find the other zeros.
31. If ? and ? are the zeros of he polynomial x
2
 – 5x + k such that ? – ? =
1, find the value of k.
32. If ??? ? are zeros of quadratic polynomial 2x
2
 + 5x + k, find the value of
k, such that ( ?? ?? ?)
2
 – ?? = 24.
33. Obtain all zeros of x
4
 – x
3
 –7x
2
 + x + 6 if 3 and 1 are zeros.
34. Find all the zeros of the polynomial 4x
4
 – 20x
3
 + 23x
2
 + 5x – 6 if two of
its zeros are 2 and 3.
X – Maths 13
35. If 
? ? ? ?
? ? 2 3 and 2 3 are  two zeroes of x
4
 – 4x
3
 – 8x
2
 + 36x – 9
find the other two zeroes.
36. What must be subtracted from 8x
4
 + 14x
3
 – 4x
2
 + 7x – 8 so that the
resulting polynomial is exactly divisible by 4x
2
 + 3x – 2.
37. When we add p(x) to 4x
4
 + 2x
3
 – 2x
2
 + x – 1 the resulting polynomial is
divisible by x
2
 + 2x – 3 find p(x).
38. Find a and f if (x
4
 + x
3
 + 8x
2
 + ax + f) is a multiple of (x
2
 + 1).
39. If the polynomial 6x
4
 + 8x
3
 + 17x
2
 + 21x + 7 is divided by 3x
2
 + 1 + 4x
then r(x) = (ax + b) find a and b.
40. Obtain all the zeroes of 2x
4
 – 2x
3
 – 7x
2
 + 3x + 6 if 
? ?
?
? ?
? ?
? ?
3
2
x are two
factors of this polynomial.
41. Find all the zeroes of x
4
 – 3x
3
 – x
2
 + 9x – 6 if – 3 and 3 are two of
its zeros.
42. If (x
3
 – 3x + 1) is one of the factors of the polynomial x
5
 – 4x
3
 + x
2
 + 3x
+ 1, find the other two factors.
43. What does the graph of the polynomial ax
2
 + bx + c represents. What
type of graph will it represent (i) for a > 0, (ii) for a < 0. What happens
if a = 0.
ANSWERS
1. b 2. a
3. a 4. b
5. a 6. c
7. b 8. b
9. a 10. d
11. a 12. a
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