CLASS IX
MATHEMATICS WORKSHEET
CHAPTER 2: POLYNOMIALS
VERY SHORT ANSWER TYPE QUESTIONS
Q1. Write an example of an algebraic expression that is not a polynomial.
Q2. Explain why the expression p(x) = √(x^3 + 1) is not a polynomial.
Q3. Find the value of the polynomial 8x^3 - 6x^2 + 2 at x = 1.
Q4. If p(x) = 6x^3 + 5x^2 – 3x + 2, find p(-1).
Q5. Find the zero(s) of the polynomial p(y) = 2y + 7.
Q6. Find the remainder when x^101 – 1 is divided by x - 1.
Q7. Determine whether xn + yn is divisible by x - y (y ≠ 0) or not.
Q8. Write the following polynomials in standard form:
i. 4y - 4y^3 + 3 - y^4
ii. 5m^3 - 2m^2 - 6m + 7
Q9. Find the integral zeroes of the following polynomials:
i. (x - 3)(x - 7)
ii. (x + 1)(3x + 2)
SHORT ANSWER TYPE QUESTIONS
Q10. If y = -1 is a zero of the polynomial q(y) = 4y^3 + ky^2 - y - 1, find the value of k.
Q11. Determine the value of m for which x^3 – 2mx^2 + 16 is divisible by x + 2.
Q12. Prove the identity: (a + b + c)^3 – a^3 – b^3 – c^3 = 3(a + b)(b + c)(c + a).
Q13. If x + 1/x = 5, find the value of x^3 + 1/x^3.
Q14. Two polynomials, x^3 + 2x^2 - 5ax - 7 and x^3 + ax^2 – 12x + 6, when divided by x + 1 and x – 2 respectively, leave remainders R1 and R2. Find the value of a in each of the following cases:
i. R1 = R2
ii. R1 + R2 = 0
iii. 2R1 + R2 = 0
Q15. If a + b + c = 9 and ab + bc + ca = 26, find the value of a^2 + b^2 + c^2.
Q16. Prove that if a + b + c = 0, then a^2bc + ab^2c + abc^2 = 3abc.
Q17. Find the zeroes of (x - 2)^2 – (x + 2)^2.
LONG ANSWER TYPE QUESTIONS
Q18. Factorize p(x) = x^4 + x^3 – 7x^2 – x + 6 using the factor theorem.
Q19. Prove that 2x^4 – 6x^3 + 3x^2 + 3x – 2 is exactly divisible by x^2 – 3x + 2.
i. By actual division
ii. Without actual division
Q20. When a polynomial p(x) = x^4 – 2x^3 + 3x^2 – ax + b is divisible by x – 1 and x + 1, the remainders are 5 and 19 respectively. Find the remainder when p(x) is divided by x – 2.
Q21. Simplify the expression: (4x^2 - 9y^2)^3 + (9y^2 - 16y^2)^3 + (16z^2 - 4x^2)^3 / (2x - 3y)^3 + (3y - 4z)^3 + (4z - 2x)^3
Q22. If x – 3 and x – 1/3 are both factors of ax^2 + 5x + b, show that a = b.
Q23. Factorize the following expressions:
i. 3(x + 2)^2 – 5(x + 2) + 2
ii. x^6 + y^6
iii. 3√(3x^3) – 5√(5y^3)
