Polynomials — Class 10 MCQs with Answers

Polynomials — important questions & MCQs with answers (Class 10 Mathematics)

14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.

1. Q1Easy

   The degree of the polynomial p(x) = 4x³ − 2x² + 7x − 5 is:

   A.1

   B.2

   C.3✓ Correct

   D.5

   Answer: C. 3

   Explanation: Degree = highest power of x with a nonzero coefficient. In 4x³ − 2x² + 7x − 5 that's x³, so the degree is 3, not the number of terms (4).

2. Q2Easy

   The degree of the polynomial 4x³ − 2x² + 5x − 7 is:

   A.3✓ Correct

   B.2

   C.1

   D.4

   Answer: A. 3

   Explanation: Degree is the highest exponent of x with a nonzero coefficient. In 4x³ − 2x² + 5x − 7 the leading term is 4x³, giving degree 3.

3. Q3Easy

   If α and β are zeroes of x² − 5x + 6, then α + β equals:

   A.6

   B.−6

   C.−5

   D.5✓ Correct

   Answer: D. 5

   Explanation: Sum of zeroes = −(coefficient of x)/(coefficient of x²) = −(−5)/1 = 5; a common slip is forgetting the negative sign, giving −5 instead.

4. Q4Easy

   When a polynomial p(x) is divided by g(x), we get a remainder r(x). The degree of r(x) is:

   A.Greater than degree of g(x)

   B.Equal to degree of g(x)

   C.Equal to degree of p(x)

   D.Less than degree of g(x)✓ Correct

   Answer: D. Less than degree of g(x)

   Explanation: Division algorithm: p(x) = g(x)·q(x) + r(x) requires deg(r) strictly less than deg(g), or r = 0; the remainder's degree can never equal or exceed the divisor's.

5. Q5Easy

   The sum of the zeroes of the polynomial 2x² − 7x + 3 is:

   A.2/7

   B.−7/2

   C.3/2

   D.7/2✓ Correct

   Answer: D. 7/2

   Explanation: For ax² + bx + c the sum of the zeroes is −b/a, which here is −(−7)/2 = 7/2.

6. Q6Easy

   A quadratic polynomial can have at most how many zeroes?

   A.2✓ Correct

   B.1

   C.3

   D.Infinite

   Answer: A. 2

   Explanation: Rule: a degree-n polynomial has at most n real zeroes. A quadratic has degree 2, so it can have at most 2 zeroes, never 3 or infinite.

7. Q7Medium

   The graph y = p(x) of a polynomial intersects the x-axis at exactly 3 points. The polynomial p(x) has:

   A.Exactly 3 real zeroes✓ Correct

   B.At most 3 zeroes

   C.At least 3 zeroes

   D.Degree exactly 3

   Answer: A. Exactly 3 real zeroes

   Explanation: Each point where the graph meets the x-axis is a real zero of the polynomial, so exactly 3 intersection points means exactly 3 real zeroes.

8. Q8Medium

   If one zero of x² − 5x + k is 2, then k equals:

   A.6✓ Correct

   B.−6

   C.10

   D.−10

   Answer: A. 6

   Explanation: Substitute the known zero into the polynomial: p(2) = 2² − 5(2) + k = 4 − 10 + k = 0, so k = 6.

9. Q9Medium

   When p(x) = x³ − 3x² + 2x − 1 is divided by (x − 2), the remainder is:

   A.−3

   B.−1✓ Correct

   C.2

   D.0

   Answer: B. −1

   Explanation: Remainder Theorem: dividing by (x − 2) leaves p(2) = 2³ − 3(2)² + 2(2) − 1 = 8 − 12 + 4 − 1 = −1.

10. Q10Medium

    If α and β are the zeroes of x² − 7x + 10, then α + β =

    A.7✓ Correct

    B.−7

    C.10

    D.−10

    Answer: A. 7

    Explanation: For ax² + bx + c, sum of zeroes = −b/a. Here a = 1, b = −7, so sum = −(−7)/1 = 7; mind the sign flip on b.

11. Q11Medium

    The zeroes of √3 x² + 10x + 7√3 are:

    A.−√3 and −7/√3✓ Correct

    B.√3 and 7/√3

    C.−√3 and 7/√3

    D.√3 and −7/√3

    Answer: A. −√3 and −7/√3

    Explanation: Discriminant = 10² − 4(√3)(7√3) = 100 − 84 = 16. x = (−10 ± 4)/(2√3), giving x = −6/(2√3) = −√3 or x = −14/(2√3) = −7/√3.

12. Q12Hard

    The positive zero of 2x² − 5x − 3 is:

    A.3✓ Correct

    B.−3

    C.1/2

    D.−1/2

    Answer: A. 3

    Explanation: 2x² − 5x − 3 = (2x + 1)(x − 3), so the zeroes are −1/2 and 3.

13. Q13Hard

    If α, β are zeroes of 2x² − 5x + 7, then the value of α²β + αβ² is:

    A.14

    B.−35/4

    C.25/4

    D.35/4✓ Correct

    Answer: D. 35/4

    Explanation: α²β + αβ² = αβ(α + β). With αβ = 7/2 and α+β = 5/2: αβ(α+β) = (7/2)(5/2) = 35/4.

14. Q14Hard

    If both (x − 2) and (x + 1) are factors of x³ − ax² + bx + 2, then a + b equals:

    A.−1

    B.1✓ Correct

    C.0

    D.2

    Answer: B. 1

    Explanation: Both roots make p(x) = 0: p(2) = 8 − 4a + 2b + 2 = 0 gives 2a − b = 5, and p(−1) = −1 − a − b + 2 = 0 simplifies directly to a + b = 1.

Polynomials — FAQs

More Class 10 Mathematics chapters

• 01Real Numbers
• 03Pair of Linear Equations in Two Variables
• 04Quadratic Equations
• 05Arithmetic Progressions
• 06Triangles
• 07Coordinate Geometry

Class 10 — other subjects

• Class 10 English chapters
• Class 10 Science chapters
• Class 10 Social Science chapters