❓ 100 MCQs — Polynomials (NCERT Class 10)
Q1
The degree of a quadratic polynomial is:
✓ Correct Answer: B
Solution: A quadratic polynomial has the form ax² + bx + c, where the highest power of x is 2.
Q2
If α and β are the zeroes of f(x) = ax² + bx + c, then α + β equals:
A c/a
B –b/a
C b/a
D –c/a
✓ Correct Answer: B
Solution: For a quadratic polynomial, the sum of zeroes = –b/a.
Q3
A cubic polynomial can have at most how many zeroes?
✓ Correct Answer: C
Solution: A polynomial of degree n can have at most n zeroes. For a cubic, n = 3.
Q4
If one zero of x² + 3x + k is 2, what is the value of k?
✓ Correct Answer: B
Solution: Substituting x = 2: 4 + 6 + k = 0 ⟹ k = –10.
Q5
The graph of a quadratic polynomial always takes the shape of a:
A Straight line
B Circle
C Parabola
D Ellipse
✓ Correct Answer: C
Solution: The graph of y = ax² + bx + c is always a parabola — opening upward when a > 0 and downward when a < 0.
Q6
If the zeroes of ax² + bx + c are reciprocals of each other, then:
A a = c
B a = b
C b = c
D a = –c
✓ Correct Answer: A
Solution: Product of zeroes = c/a. If the zeroes are reciprocals, their product = 1, so c/a = 1, meaning c = a.
Q7
The zeroes of the polynomial x² – 3 are:
A 3, –3
B √3, –√3
C 9, –9
D √3, 3
✓ Correct Answer: B
Solution: x² = 3 ⟹ x = ±√3.
Q8
What is the value of p(x) = 5x² – 3x + 7 at x = 1?
✓ Correct Answer: A
Solution: p(1) = 5 – 3 + 7 = 9.
Q9
A polynomial of degree 3 is called a:
A Linear polynomial
B Quadratic polynomial
C Cubic polynomial
D Biquadratic polynomial
✓ Correct Answer: C
Solution: A polynomial with degree 3 is called a cubic polynomial.
Q10
If the graph crosses the x-axis at 3 points, the number of zeroes is:
✓ Correct Answer: D
Solution: The number of zeroes equals the number of times the graph meets the x-axis.
Q11
If ax² – 6x – 6 has a product of zeroes equal to 4, then a =
✓ Correct Answer: A
Solution: Product of zeroes = c/a = –6/a = 4 ⟹ a = –3/2.
Q12
Which of the following is a polynomial?
A x² + 1/x
B √x + 2
C x² + 3x + 4
D x^(3/2) + 5
✓ Correct Answer: C
Solution: In a polynomial, all exponents must be non-negative integers. Only x² + 3x + 4 satisfies this condition.
Q13
A linear polynomial has exactly how many zeroes?
A None
B One
C Two
D Infinitely many
✓ Correct Answer: B
Solution: A degree-1 polynomial has exactly one zero.
Q14
If α, β are zeroes of x² – 5x + k and α – β = 1, then k =
✓ Correct Answer: B
Solution: α + β = 5 and α – β = 1 ⟹ α = 3, β = 2. Therefore k = αβ = 6.
Q15
If the sum of zeroes is 0 and the product is √5, the polynomial is:
A x² – √5
B x² + √5
C x² – x + √5
D x² + x – √5
✓ Correct Answer: B
Solution: p(x) = x² – (sum)x + product = x² – 0·x + √5 = x² + √5.
Q16
If the sum of zeroes of kx² + 2x + 3k equals their product, then k =
A –2/3
B 2/3
C 1/3
D –1/3
✓ Correct Answer: A
Solution: Sum = –2/k, product = 3. Setting –2/k = 3 ⟹ k = –2/3.
Q17
The quadratic polynomial with zeroes –3 and 4 is:
A x² – x + 12
B x² + x + 12
C x² – x – 12
D 2x² + 2x – 24
✓ Correct Answer: C
Solution: Sum = 1, product = –12. Polynomial = x² – x – 12.
Q18
If x² + (a+1)x + b has zeroes 2 and –3, then:
A a = –7, b = –1
B a = 5, b = –1
C a = 2, b = –6
D a = 0, b = –6
✓ Correct Answer: D
Solution: Sum: –(a+1) = –1 ⟹ a = 0. Product: b = 2 × (–3) = –6.
Q19
The zeroes of x² – 2x – 8 are:
A –2, 4
B 2, –4
C 2, 4
D –2, –4
✓ Correct Answer: A
Solution: (x – 4)(x + 2) = 0 ⟹ x = 4, x = –2.
Q20
In the Division Algorithm p(x) = g(x)×q(x) + r(x), the degree of r(x) must be:
A Equal to degree of g(x)
B Greater than degree of g(x)
C Less than degree of g(x)
D Always zero
✓ Correct Answer: C
Solution: The degree of the remainder must always be strictly less than the degree of the divisor.
Q21
A polynomial of degree n can intersect the x-axis at most how many times?
✓ Correct Answer: C
Solution: A degree-n polynomial has at most n real zeroes, so it can meet the x-axis at most n times.
Q22
If α and β are zeroes of 2x² + 5x – 10, then αβ =
✓ Correct Answer: B
Solution: αβ = c/a = –10/2 = –5.
Q23
The zero of the linear polynomial ax + b is:
A b/a
B –b/a
C a/b
D –a/b
✓ Correct Answer: B
Solution: ax + b = 0 ⟹ x = –b/a.
Q24
An example of a quadratic polynomial with only one (repeated) zero is:
A x² – 4
B x² + 4x + 4
C x² + 1
D x² – 2x
✓ Correct Answer: B
Solution: x² + 4x + 4 = (x+2)², so the only zero is –2 (repeated).
Q25
If the graph does not cross the x-axis at all, the number of real zeroes is:
A 1
B 0
C 2
D Infinitely many
✓ Correct Answer: B
Solution: No x-intercepts means no real zeroes.
Q26
The product of all three zeroes of the cubic ax³ + bx² + cx + d is:
A –b/a
B c/a
C –d/a
D d/a
✓ Correct Answer: C
Solution: For a cubic polynomial, the product of all three zeroes = –d/a.
Q27
If α, β are zeroes of x² – kx + 6 and α + β = 5, then k =
✓ Correct Answer: B
Solution: Sum of zeroes = k. Since α + β = 5, k = 5.
Q28
If the zeroes of p(x) = x² – 1 are α and β, then α + β =
✓ Correct Answer: C
Solution: The coefficient of x is 0, so sum = –0/1 = 0.
Q29
The zeroes of p(x) = x² – 27 are:
✓ Correct Answer: A
Solution: x² = 27 ⟹ x = ±√27 = ±3√3.
Q30
The degree of a non-zero constant polynomial is:
A 1
B 0
C Not defined
D Any natural number
✓ Correct Answer: B
Solution: A constant c can be written as cx⁰, so its degree is 0.
Q31
The degree of the zero polynomial is:
A 0
B 1
C Not defined
D Infinite
✓ Correct Answer: C
Solution: By convention, the degree of the zero polynomial is not defined.
Q32
If α, β are zeroes of 4x² + 3x + 7, then 1/α + 1/β =
A 3/7
B –3/7
C 7/3
D –7/3
✓ Correct Answer: B
Solution: (α+β)/αβ = (–3/4)/(7/4) = –3/7.
Q33
If the sum of zeroes of x² – (k+6)x + 2(2k–1) is half their product, then k =
✓ Correct Answer: B
Solution: k + 6 = 2k – 1 ⟹ k = 7.
Q34
The quadratic polynomial with zeroes √2 and –√2 is:
A x² – 2
B x² + 2
C x² – √2
D x² + √2
✓ Correct Answer: A
Solution: Product = –2, sum = 0. Polynomial = x² – 2.
Q35
If α, β are zeroes of x² + x + 1, then α² + β² =
✓ Correct Answer: B
Solution: (α+β)² – 2αβ = (–1)² – 2(1) = 1 – 2 = –1.
Q36
The parabola y = –3x² + 2x – 1 opens:
A Upward
B Downward
C Rightward
D Leftward
✓ Correct Answer: B
Solution: The coefficient of x² is negative (–3), so the parabola opens downward.
Q37
In p(x) = ax² + bx + c, the ratio c/a represents:
A Sum of zeroes
B Product of zeroes
C Difference of zeroes
D Nothing specific
✓ Correct Answer: B
Solution: c/a represents the product of the zeroes of a quadratic polynomial.
Q38
How many polynomials can have zeroes –2 and 5?
A 1
B 2
C 3
D More than one
✓ Correct Answer: D
Solution: k(x+2)(x–5) has these zeroes for any non-zero constant k, so infinitely many polynomials are possible.
Q39
If both zeroes of ax² + bx + c are positive, then in ax² + bx + c:
A a, b, c have the same sign
B c and a have the same sign; b has the opposite sign
C b and c have the same sign; a has the opposite sign
D a and b have the same sign; c has the opposite sign
✓ Correct Answer: B
Solution: Sum > 0 ⟹ –b/a > 0, so b and a have opposite signs. Product > 0 ⟹ c/a > 0, so c and a have the same sign.
Q40
If one zero of x³ + ax² + bx + c is –1, the product of the remaining two zeroes is:
A b – a + 1
B b – a – 1
C a – b + 1
D a – b – 1
✓ Correct Answer: A
Solution: p(–1) = 0 gives c = 1 – a + b. Product of remaining two zeroes = c = b – a + 1.
Q41
If x² – 6x + k = 0 and 3α + 2β = 20, where α, β are zeroes, then k =
✓ Correct Answer: A
Solution: α + β = 6 and 3α + 2β = 20 ⟹ α = 8, β = –2. k = αβ = –16.
Q42
A quadratic polynomial cannot intersect the x-axis at three points because:
A Its graph is a parabola
B It can have at most 2 real zeroes
C Its graph is a straight line
D Its degree is 1
✓ Correct Answer: B
Solution: A degree-2 polynomial can have at most 2 real zeroes, so its graph cannot cut the x-axis at 3 points.
Q43
The zeroes of x² + 99x + 127 are:
A Both positive
B Both negative
C One positive, one negative
D Both equal
✓ Correct Answer: B
Solution: Sum = –99 (negative), product = 127 (positive). Both zeroes are negative.
Q44
For x² + kx + k (k ≠ 0), the zeroes:
A Cannot both be positive
B Cannot both be negative
C Are always distinct
D Are always equal
✓ Correct Answer: A
Solution: If k > 0, sum = –k < 0, ruling out both positive. If k < 0, product = k < 0, meaning one is positive and one negative. Either way, both cannot be positive.
Q45
The degree of a linear polynomial is:
✓ Correct Answer: B
Solution: Linear means degree 1. Example: 3x + 2.
Q46
If α, β are zeroes of x² – 4x + 1, then 1/α + 1/β – αβ =
✓ Correct Answer: A
Solution: (α+β)/αβ – αβ = 4/1 – 1 = 3.
Q47
The zeroes of a polynomial p(x) are the x-values where the graph:
A Crosses the y-axis
B Reaches its end point
C Crosses the x-axis
D Reaches its vertex
✓ Correct Answer: C
Solution: This is the very definition of a zero — the x-value at which p(x) = 0, i.e., where the graph meets the x-axis.
Q48
The general form of a cubic polynomial with zeroes α, β, γ is:
A (x–α)(x–β)(x–γ)
B k(x–α)(x–β)(x–γ)
C x³–(α+β+γ)x²
D None of the above
✓ Correct Answer: B
Solution: The general form always includes an arbitrary non-zero constant k.
Q49
If p(x) = x² – 5x + 6, then p(2) =
✓ Correct Answer: A
Solution: p(2) = 4 – 10 + 6 = 0.
Q50
The polynomial whose zeroes are the reciprocals of those of ax² + bx + c is:
A cx² + bx + a
B ax² – bx + c
C cx² – bx + a
D bx² + ax + c
✓ Correct Answer: A
Solution: Replace x with 1/x in ax² + bx + c and simplify to get cx² + bx + a.
Q51
If the sum of zeroes is –1/4 and the product is 1/4, the polynomial is:
A 4x² + x + 1
B 4x² – x + 1
C x² + 4x + 1
D x² – 4x + 1
✓ Correct Answer: A
Solution: p(x) = x² + (1/4)x + 1/4. Multiplying through by 4 gives 4x² + x + 1.
Q52
If the zeroes of x³ – 3x² + x + 1 are (a–d), a, (a+d) in AP, then a =
✓ Correct Answer: A
Solution: Sum of zeroes = 3a = 3 ⟹ a = 1.
Q53
Which of the following is NOT a polynomial?
A x² + 2x
B 3
C 1/(x–1)
D x³
✓ Correct Answer: C
Solution: 1/(x–1) has a variable in the denominator, giving a negative exponent — that disqualifies it as a polynomial.
Q54
How many zeroes does p(x) = x² + 2x + 1 have?
A No real zeroes
B Two distinct zeroes
C Two equal zeroes
D One real zero
✓ Correct Answer: C
Solution: (x+1)² = 0 ⟹ zero is –1 (repeated twice).
Q55
In x² – 8x + k, if one zero is three times the other, then k =
✓ Correct Answer: A
Solution: Let zeroes be α and 3α. Then 4α = 8 ⟹ α = 2, 3α = 6. k = 2 × 6 = 12.
Q56
A cubic polynomial always has at least how many real zeroes?
✓ Correct Answer: B
Solution: A polynomial of odd degree always has at least one real zero.
Q57
The quadratic polynomial with zeroes 1 and –2 is:
A x² + x – 2
B x² – x – 2
C x² + x + 2
D x² – x + 2
✓ Correct Answer: A
Solution: Sum = –1, product = –2. Polynomial = x² – (–1)x + (–2) = x² + x – 2.
Q58
The degree of the product of a degree-2 and a degree-3 polynomial is:
✓ Correct Answer: A
Solution: Degree of product = sum of degrees = 2 + 3 = 5.
Q59
If α, β are zeroes of x² – p(x+1) – c, then (α+1)(β+1) =
A c
B 1 – c
C c – 1
D 1 + c
✓ Correct Answer: B
Solution: Expanding: αβ + (α+β) + 1 = 1 – c.
Q60
If (x + a) is a factor of 2x² + 2ax + 5x + 10, then a =
✓ Correct Answer: A
Solution: By the Factor Theorem, p(–a) = 0. Solving gives a = 2.
Q61
If the graph touches the x-axis once but does not cross it, the number of zeroes is:
✓ Correct Answer: B
Solution: Touching (not crossing) the x-axis gives one repeated zero.
Q62
If x² – kx + 6 has integer zeroes, k could be:
✓ Correct Answer: C
Solution: Integer factor pairs of 6 whose sum is 5: (2, 3). So k = 5.
Q63
The degree of (x+1)(x²+x–1) is:
✓ Correct Answer: B
Solution: The highest power in the expanded form is x³.
Q64
If α, β are zeroes of 3x² + 8x + 2, then α² + β² =
A 52/9
B 64/9
C 40/9
D 44/9
✓ Correct Answer: A
Solution: (α+β)² – 2αβ = 64/9 – 12/9 = 52/9.
Q65
The zero of p(x) = 2x – 5 is:
✓ Correct Answer: B
Solution: 2x = 5 ⟹ x = 5/2.
Q66
If α, β are zeroes of x² + ax – b, the polynomial whose zeroes are 1/α and 1/β is:
A bx² – ax – 1
B bx² + ax – 1
C x² + ax – b
D None of the above
✓ Correct Answer: A
Solution: Replacing x with 1/x in x² + ax – b and simplifying gives bx² – ax – 1 = 0.
Q67
Can a quadratic polynomial have 3 zeroes?
A Yes
B No
C Only the zero polynomial can
D Sometimes
✓ Correct Answer: B
Solution: A degree-2 polynomial can have at most 2 real zeroes.
Q68
If the zeroes of x³ – 3x² + x + 1 are (a–b), a, (a+b), then b =
✓ Correct Answer: A
Solution: a = 1. Then 1 – b² = –1 ⟹ b = √2.
Q69
The zeroes of x² – 15 are:
A ±15
B ±√15
C 15, 0
D √15, 15
✓ Correct Answer: B
Solution: x = ±√15.
Q70
The sum of zeroes of 3x² + 5x – 2 is:
A 5/3
B –5/3
C 2/3
D –2/3
✓ Correct Answer: B
Solution: –b/a = –5/3.
Q71
The product of zeroes of 3x² + 5x – 2 is:
A 2/3
B –2/3
C 5/3
D –5/3
✓ Correct Answer: B
Solution: c/a = –2/3.
Q72
If p(x) = x – 3, then p(x) + p(–x) =
✓ Correct Answer: B
Solution: (x–3) + (–x–3) = –6.
Q73
The zeroes of x² – kx + k = 0 are equal when k =
✓ Correct Answer: C
Solution: Discriminant = k² – 4k = 0 ⟹ k(k–4) = 0 ⟹ k = 0 or 4.
Q74
A non-zero constant polynomial k (k ≠ 0) has how many zeroes?
A 0
B 1
C Infinitely many
D k
✓ Correct Answer: A
Solution: Its graph is a horizontal line that never touches the x-axis, so it has no zeroes.
Q75
If α, β are zeroes of x² – 1, then α² + β² =
✓ Correct Answer: B
Solution: Zeroes are 1 and –1. 1² + (–1)² = 2.
Q76
The zeroes of x² – 5x are:
A 0, 5
B 5, 5
C 0, 0
D –5, 5
✓ Correct Answer: A
Solution: x(x–5) = 0 ⟹ x = 0 or x = 5.
Q77
The degree of x⁴ – x³ + x² – 1 is:
✓ Correct Answer: D
Solution: The highest power of x is 4.
Q78
If a and c in ax² + bx + c have opposite signs, the zeroes are:
A Both positive
B Both negative
C One positive, one negative
D No real zeroes
✓ Correct Answer: C
Solution: c/a is negative ⟹ product of zeroes is negative ⟹ one zero is positive and the other is negative.
Q79
If α, β are zeroes of x² + x + 1, then 1/α + 1/β =
A 1
B –1
C 0
D Not defined
✓ Correct Answer: B
Solution: (α+β)/αβ = –1/1 = –1.
Q80
If f(x) = 2x² – 3x + 5, then f(–1) =
✓ Correct Answer: A
Solution: 2(1) – 3(–1) + 5 = 2 + 3 + 5 = 10.
Q81
The zeroes of x² – √2x – 4 are:
A 2√2, –√2
B –2√2, √2
C √2, √2
D 2, 2
✓ Correct Answer: A
Solution: Sum = √2, product = –4. The zeroes are 2√2 and –√2.
Q82
The quadratic polynomial with zeroes 0 and 5 is:
A x² – 5x
B x² + 5x
C x² – 5
D x² + 5
✓ Correct Answer: A
Solution: x(x–5) = x² – 5x.
Q83
If kx² + 4x + 3k has a sum of zeroes equal to 2, then k =
✓ Correct Answer: B
Solution: –4/k = 2 ⟹ k = –2.
Q84
If y = f(x) has a graph parallel to the x-axis, the number of zeroes is:
A 0
B 1
C Infinitely many
D None of the above
✓ Correct Answer: A
Solution: A line parallel to the x-axis (and not coinciding with it) never intersects it, so no real zeroes.
Q85
The product of zeroes of x³ – 4x is:
✓ Correct Answer: C
Solution: The constant term d = 0, so product = –d/a = 0.
Q86
If α, β are zeroes of x² – 4, then α³ + β³ =
✓ Correct Answer: A
Solution: Zeroes are 2 and –2. 8 + (–8) = 0.
Q87
If x² – kx + 5 has a sum of zeroes equal to 7, then k =
✓ Correct Answer: B
Solution: Sum = k. Given sum = 7, so k = 7.
Q88
By the Factor Theorem, 'a' is a zero of p(x) if and only if ______ is a factor:
A x + a
B x – a
C ax
D x/a
✓ Correct Answer: B
Solution: The Factor Theorem states that (x – a) is a factor of p(x) if and only if p(a) = 0.
Q89
If p(x) = x² + 5x + 4, the zeroes of p(x) are:
A 1, 4
B –1, –4
C 1, –4
D –1, 4
✓ Correct Answer: B
Solution: (x+1)(x+4) = 0 ⟹ x = –1 and x = –4.
Q90
If α, β are zeroes of ax² + bx + c, then α²β + αβ² =
A –bc/a²
B bc/a²
C –ab/c²
D None of the above
✓ Correct Answer: A
Solution: αβ(α+β) = (c/a)(–b/a) = –bc/a².
Q91
Where does the graph of y = x² – x – 6 cross the y-axis?
A (0, –6)
B (–6, 0)
C (0, 6)
D (6, 0)
✓ Correct Answer: A
Solution: At x = 0: y = 0 – 0 – 6 = –6. The y-intercept is (0, –6).
Q92
If 3x² + 5x + k has one zero equal to the reciprocal of the other, then k =
✓ Correct Answer: A
Solution: Product of zeroes = 1 ⟹ k/3 = 1 ⟹ k = 3.
Q93
The zeroes of x² – 2x are:
A 0, 2
B 1, 1
C –2, 0
D 2, 2
✓ Correct Answer: A
Solution: x(x–2) = 0 ⟹ x = 0 or x = 2.
Q94
The degree of x(x+1)(x+2) is:
✓ Correct Answer: C
Solution: Expanding gives x³ + 3x² + 2x, so the degree is 3.
Q95
If α, β are zeroes of x² – x – 2, then a possible value of α – β is:
✓ Correct Answer: C
Solution: Zeroes are 2 and –1. α – β = 2 – (–1) = 3.
Q96
A quadratic polynomial with sum 0 and one zero equal to 3 is:
A x² – 9
B x² + 9
C x² – 3
D x² + 3
✓ Correct Answer: A
Solution: Sum = 0 means the other zero is –3. Product = –9. Polynomial = x² – 9.
Q97
If α, β are zeroes of x² – 5x + 6, then α/β + β/α =
✓ Correct Answer: A
Solution: [(α+β)² – 2αβ]/αβ = [25 – 12]/6 = 13/6.
Q98
Which of the following is a linear polynomial?
A x + 2
B x² + 2
C 1/x
D 2
✓ Correct Answer: A
Solution: x + 2 has degree 1, making it a linear polynomial.
Q99
The number of zeroes of p(x) = (x–1)(x–2)(x–3)(x–4) is:
✓ Correct Answer: D
Solution: There are 4 distinct linear factors, giving 4 distinct real zeroes.
Q100
If α, β are zeroes of x² – 4x + 3, then 1/α + 1/β =
✓ Correct Answer: B
Solution: (α+β)/αβ = 4/3.