Class 10 Mathematics: Polynomials important questions

Polynomials is a chapter of Class 10 Mathematics for the Andhra Pradesh State Board. These 19 questions are marked Very Important, Important or Practice, and each answer has been checked by our team. Use them to revise, not as a prediction of what will come in the exam.

1. Practice · 1 marks

   1. Statement A: A cubic polynomial has at most 3 zeroes. Statement B: The zero of the linear polynomial x − 2 is −2. i) Both true ii) A true, B false iii) A false, B true iv) Both false

   ii) A is true (degree 3 means at most 3 zeroes). B is false: x − 2 = 0 gives x = 2.

2. Important · 4 marks

   2. Find the zeroes of x² + 7x + 10 and verify the relation between the zeroes and the coefficients.

   x² + 7x + 10 = (x + 2)(x + 5), so the zeroes are −2 and −5. Sum = −2 + (−5) = −7 = −b/a = −7/1. ✓ Product = (−2)(−5) = 10 = c/a = 10/1. ✓

3. Very Important · 8 marks

   3. Draw the graph of p(x) = x² − 2x − 8 and find its zeroes.

   Table of values: x: −3, −2, −1, 0, 1, 2, 3, 4, 5 y: 7, 0, −5, −8, −9, −8, −5, 0, 7 Plot the points and join them with a smooth curve: a parabola opening upwards with lowest point (1, −9). It cuts the x-axis at x = −2 and x = 4, so the zeroes are −2 and 4. (Check: x² − 2x − 8 = (x + 2)(x − 4).)

4. Practice · 1 marks

   4. Assertion: The sum of the zeroes of 2x² + 3x − 4 is −3/2. Reason: The sum of the zeroes of ax² + bx + c is c/a. A) Both are true and the Reason supports the Assertion. B) Both are true but the Reason does not support the Assertion. C) The Assertion is true, but the Reason is false. D) The Assertion is false, but the Reason is true.

   C) Sum of zeroes = −b/a = −3/2, so the Assertion is true. The Reason is false: c/a is the product of the zeroes, not the sum.

5. Practice · 1 marks

   5. The zero of the polynomial ax + b is A) b/a B) a/b C) −a/b D) −b/a

   D) −b/a, since ax + b = 0 gives x = −b/a.

6. Important · 2 marks

   6. Find a quadratic polynomial whose sum and product of zeroes are 3 and 2 respectively.

   x² − 3x + 2 (or any non-zero multiple).

7. Practice · 1 marks

   7. The coefficient of x in p(x) = 7x³ − 6x² + 5x + 8 is A) 7 B) −6 C) 5 D) −5

   C) 5

8. Practice · 1 marks

   8. Write a polynomial of degree 4 whose constant term is −6.

   For example, x⁴ + 2x² − 6. Any polynomial whose highest power of x is 4 and whose constant term is −6 is correct.

9. Important · 2 marks

   9. Create a quadratic polynomial whose sum and product of zeroes are 3 and −2 respectively.

   x² − (sum)x + (product) = x² − 3x − 2 (or any non-zero multiple).

10. Important · 4 marks

    10. The graph of a polynomial is a parabola opening upwards that cuts the x-axis at x = −3 and x = 4. 1) What is the shape of the graph? 2) How many zeroes does it have? 3) What are the zeroes? 4) Find the sum of the zeroes.

    1) A parabola (opening upwards). 2) Two zeroes. 3) −3 and 4. 4) Sum = −3 + 4 = 1.

11. Practice · 1 marks

    11. A polynomial is an algebraic expression in which the exponent of every variable is a A) integer B) whole number C) rational number D) real number

    B) whole number (0, 1, 2, 3, …). Negative or fractional powers are not allowed in a polynomial.

12. Practice · 1 marks

    12. Create a cubic polynomial which is a binomial.

    For example, x³ + 2. It has degree 3 (cubic) and exactly two terms (binomial).

13. Important · 2 marks

    13. Create a quadratic polynomial whose sum and product of zeroes are equal.

    Take sum = product = 4 (zeroes 2 and 2): x² − 4x + 4. In general, x² − kx + k for any k has equal sum and product of zeroes.

14. Practice · 1 marks

    14. Which of these polynomials has zeroes 1 and 2? A) x² − 3x + 2 B) x² + 3x + 2 C) x² − 2x + 3 D) x² + 2x − 3

    A) x² − 3x + 2 A polynomial with zeroes 1 and 2 is (x − 1)(x − 2) = x² − 3x + 2. Check: sum of zeroes = 3 and product = 2.

15. Important · 2 marks

    15. Form a quadratic polynomial whose zeroes are α and β, if α + β = 6 and αβ = 4.

    A quadratic polynomial with sum of zeroes S and product P is k(x² − Sx + P). So the polynomial is x² − 6x + 4 (or any non-zero multiple of it).

16. Important · 4 marks

    16. The graph of a polynomial y = ax² + bx + c is a parabola opening upwards that cuts the x-axis at x = −1 and x = 3. i) How many zeroes does the polynomial have? ii) What are the zeroes? iii) What is the value of −b/a? iv) What is the product of its zeroes?

    i) 2, because the graph meets the x-axis at two points. ii) −1 and 3. iii) −b/a = sum of zeroes = −1 + 3 = 2. iv) Product of zeroes = (−1)(3) = −3.

17. Practice · 1 marks

    17. Create a quadratic polynomial one of whose zeroes is 2 + √3.

    Irrational zeroes of a polynomial with rational coefficients come in pairs, so the other zero is 2 − √3. Sum = 4, product = (2 + √3)(2 − √3) = 4 − 3 = 1. Polynomial: x² − 4x + 1.

18. Important · 2 marks

    18. Write two different linear polynomials in x whose zero is −4.

    x + 4 and 2x + 8. (Any k(x + 4) with k ≠ 0 has zero −4.)

19. Important · 4 marks

    19. The graph of a quadratic polynomial is a parabola opening upwards that cuts the x-axis at x = 2 and x = 3. 1) Write the zeroes. 2) Find the sum of the zeroes. 3) Find the product of the zeroes. 4) Name the shape of the graph.

    1) The zeroes are 2 and 3. 2) Sum = 2 + 3 = 5. 3) Product = 2 × 3 = 6. 4) The graph is a parabola (opening upwards). The polynomial is k(x² − 5x + 6).