Chapter 2: Polynomials

Class 10 Mathematics · MCQs with answers

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What is the degree of a quadratic polynomial?
1. (A) 3
2. (B) 2
3. (C) 1
4. (D) 0
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Correct answer: (B) 2

Explanation: The general form of a quadratic polynomial is ax² + bx + c (a ≠ 0). The highest power of x is 2.

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What is the zero of the polynomial p(x) = 2x + 3?
1. (A) −3/2
2. (B) −2/3
3. (C) 2/3
4. (D) 3/2
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Correct answer: (A) −3/2

Explanation: Setting 2x + 3 = 0 gives x = −3/2. At this value p(x) = 0, so it is the zero.

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What are the zeroes of x² − 2x − 8?
1. (A) −4 and 2
2. (B) 8 and −1
3. (C) −8 and 1
4. (D) 4 and −2
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Correct answer: (D) 4 and −2

Explanation: x² − 2x − 8 = (x − 4)(x + 2). So the polynomial is zero at x = 4 or x = −2.

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What is the sum of the zeroes of 6x² − 7x − 3?
1. (A) −7/6
2. (B) −1/2
3. (C) 1/2
4. (D) 7/6
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Correct answer: (D) 7/6

Explanation: For ax² + bx + c, the sum of the zeroes is −b/a. Here a = 6 and b = −7, so the sum is 7/6. (The zeroes are 3/2 and −1/3.)

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What is the product of the zeroes of 3x² − x − 4?
1. (A) 4/3
2. (B) 1/3
3. (C) −1/3
4. (D) −4/3
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Correct answer: (D) −4/3

Explanation: The product of the zeroes is c/a = −4/3. (The zeroes are 4/3 and −1, and 4/3 × (−1) = −4/3.)

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Which quadratic polynomial has zeroes whose sum is −3 and product is 2?
1. (A) x² − 3x + 2
2. (B) x² + 3x − 2
3. (C) x² − 2x + 3
4. (D) x² + 3x + 2
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Correct answer: (D) x² + 3x + 2

Explanation: Polynomial = x² − (sum)x + product = x² − (−3)x + 2 = x² + 3x + 2. Its zeroes are −1 and −2.

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Which quadratic polynomial has zeroes 1/4 and −1?
1. (A) 4x² − 3x − 1
2. (B) 4x² + 3x − 1
3. (C) 4x² + 3x + 1
4. (D) x² + 3x − 4
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Correct answer: (B) 4x² + 3x − 1

Explanation: Sum = 1/4 − 1 = −3/4 and product = −1/4. Polynomial = x² + (3/4)x − 1/4; multiplying by 4 gives 4x² + 3x − 1.

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The graph of a polynomial y = p(x) cuts the x-axis at exactly 3 points. How many zeroes does p(x) have?
1. (A) 0
2. (B) 1
3. (C) 2
4. (D) 3
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Correct answer: (D) 3

Explanation: A polynomial has as many zeroes as the points where its graph cuts or touches the x-axis. Here there are 3 points.

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If the graph of y = p(x) does not meet the x-axis anywhere, how many zeroes does p(x) have?
1. (A) 1
2. (B) 2
3. (C) 0
4. (D) Infinitely many
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Correct answer: (C) 0

Explanation: Zeroes are where the graph meets the x-axis. This graph never meets it, so there is no real zero.

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At most how many zeroes can a quadratic polynomial have?
1. (A) 1
2. (B) 2
3. (C) 3
4. (D) Infinitely many
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Correct answer: (B) 2

Explanation: A polynomial of degree n has at most n zeroes. A quadratic has degree 2, so at most 2 zeroes.

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If 2 is a zero of the polynomial x² − 5x + k, what is the value of k?
1. (A) 10
2. (B) 6
3. (C) 4
4. (D) −6
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Correct answer: (B) 6

Explanation: Putting x = 2 gives 4 − 10 + k = 0, so k = 6.

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If α and β are the zeroes of x² − 6x + 8, what is α² + β²?
1. (A) 20
2. (B) 28
3. (C) 36
4. (D) 52
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Correct answer: (A) 20

Explanation: α + β = 6 and αβ = 8. α² + β² = (α + β)² − 2αβ = 36 − 16 = 20. (The zeroes are 2 and 4: 4 + 16 = 20.)

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If α and β are the zeroes of x² − 5x + 4, what is 1/α + 1/β?
1. (A) 5/4
2. (B) 4/5
3. (C) 1/4
4. (D) −5/4
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Correct answer: (A) 5/4

Explanation: 1/α + 1/β = (α + β)/(αβ) = 5/4. (The zeroes are 1 and 4: 1 + 1/4 = 5/4.)

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What are the zeroes of the polynomial 4u² + 8u?
1. (A) 0 and 2
2. (B) −2 and 2
3. (C) 0 and −8
4. (D) 0 and −2
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Correct answer: (D) 0 and −2

Explanation: 4u² + 8u = 4u(u + 2). It is zero when u = 0 or u = −2.

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What are the zeroes of the polynomial t² − 15?
1. (A) 15 and −15
2. (B) √15 and −√15
3. (C) 15 only
4. (D) No real zeroes
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Correct answer: (B) √15 and −√15

Explanation: t² − 15 = (t − √15)(t + √15), so t = √15 or t = −√15.

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If the two zeroes of ax² + bx + c are equal in size but opposite in sign, which is true?
1. (A) b = 0
2. (B) a = 0
3. (C) c = 0
4. (D) a = c
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Correct answer: (A) b = 0

Explanation: If the zeroes are k and −k, their sum is 0. Sum = −b/a = 0, so b = 0.

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If the zeroes of ax² + bx + c are reciprocals of each other, which is true?
1. (A) b = 0
2. (B) c = 0
3. (C) c = a
4. (D) a = −c
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Correct answer: (C) c = a

Explanation: If the zeroes are k and 1/k, their product is 1. Product = c/a = 1, so c = a.

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When x³ − 3x² + 5x − 3 is divided by x² − 2, what are the quotient and remainder?
1. (A) Quotient x + 3, remainder 7x − 9
2. (B) Quotient x − 3, remainder 7x − 9
3. (C) Quotient x − 3, remainder 7x + 9
4. (D) Quotient x − 3, remainder −7x + 9
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Correct answer: (B) Quotient x − 3, remainder 7x − 9

Explanation: Check: (x² − 2)(x − 3) + (7x − 9) = x³ − 3x² − 2x + 6 + 7x − 9 = x³ − 3x² + 5x − 3.

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In the division algorithm for polynomials, p(x) = g(x) × q(x) + r(x), what condition does r(x) satisfy?
1. (A) Degree of r(x) > degree of g(x)
2. (B) Degree of r(x) = degree of p(x)
3. (C) r(x) is always a constant
4. (D) r(x) = 0 or degree of r(x) < degree of g(x)
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Correct answer: (D) r(x) = 0 or degree of r(x) < degree of g(x)

Explanation: The remainder is either zero or has a lower degree than the divisor g(x).

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What is the sum of the zeroes of the cubic polynomial ax³ + bx² + cx + d?
1. (A) b/a
2. (B) c/a
3. (C) −b/a
4. (D) −d/a
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Correct answer: (C) −b/a

Explanation: For a cubic: sum of zeroes = −b/a, sum of products taken two at a time = c/a, and product of all three = −d/a.

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What is the product of the three zeroes of 2x³ − 5x² − 14x + 8?
1. (A) 4
2. (B) 5/2
3. (C) −4
4. (D) −7
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Correct answer: (C) −4

Explanation: Product of the three zeroes = −d/a = −8/2 = −4. (The zeroes are 4, −2 and 1/2: 4 × (−2) × 1/2 = −4.)

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Which of these expressions is not a polynomial?
1. (A) x² + 3x
2. (B) 5
3. (C) 2x³ − 1
4. (D) x + 1/x
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Correct answer: (D) x + 1/x

Explanation: In a polynomial, the powers of the variable are whole numbers (0, 1, 2, …). 1/x = x⁻¹ has a negative power, so x + 1/x is not a polynomial.

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Which of these values is a zero of x² − 3x + 2?
1. (A) 3
2. (B) 2
3. (C) 0
4. (D) −1
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Correct answer: (B) 2

Explanation: Putting x = 2 gives 4 − 6 + 2 = 0. The polynomial is not zero at the other values. (Its other zero is 1.)

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What is the zero of the linear polynomial ax + b (a ≠ 0)?
1. (A) −b/a
2. (B) b/a
3. (C) −a/b
4. (D) a/b
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Correct answer: (A) −b/a

Explanation: ax + b = 0 gives x = −b/a. A linear polynomial has exactly one zero.

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Assertion (A): The polynomial x² + 4 has no real zeroes. Reason (R): The graph of y = x² + 4 always stays above the x-axis and never cuts it.
1. (A) Both A and R are true, and R is the correct explanation of A.
2. (B) Both A and R are true, but R is not the correct explanation of A.
3. (C) A is true, but R is false.
4. (D) A is false, but R is true.
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Correct answer: (A) Both A and R are true, and R is the correct explanation of A.

Explanation: x² ≥ 0, so x² + 4 ≥ 4 and is never zero. The graph never cuts the x-axis, so R correctly explains A.

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