Polynomials in Mathematics

Updated on August 22, 2025 | By Learnzy Academy

A polynomial is an algebraic expression consisting of variables (also called indeterminates), coefficients, and exponents, that are combined using addition, subtraction, and multiplication. The general form of a polynomial is:

P(x)=an​xn+an-1xn-1+ ⋯+a​1x+ a0

where:

  • a0, a1,… are constants called coefficients,

  • x is the variable, and

  • n is a non-negative integer called the degree of the polynomial.

Polynomials are fundamental in algebra and are used to represent mathematical models, solve equations, and describe various relationships in science and engineering.

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List of question on "Polynomials"

  1. If (x – 1/x) = 4, then evaluate (x² + 1/x²) and (x⁴ + 1/x⁴).
  2. Factorise 64m³ – 343n³
  3. Factorise 8a³ + b³ + 12a²b + 6ab²
  4. Without actual division, prove that 2x^4 – 5x^3 + 2x^2 – x + 2 is divisible by x^2 – 3x + 2.
  5. Evaluate (102)^3 using a suitable identity.
  6. Factorise x^2 – 1 – 2a – a^2.
  7. Check whether (7 + 3x) is a factor of (3x^3 + 7x).
  8. Find the values of a and b so that (2x^3 + a x^2 + x + b) has (x + 2) and (2x – 1) as factors.
  9. Find the value of x^3 + y^3 + z^3 – 3xyz if x + y + z = 15 and x^2 + y^2 + z^2 = 83
  10. Calculate the perimeter of a rectangle whose area is 25x^2 - 35x + 12
  11. Give an example of a monomial and a binomial having degrees of 82 and 99, respectively.
  12. The value of 104 × 96 is
  13. The value of 5.63 × 5.63 + 11.26 × 2.37 + 2.37 × 2.37 is
  14. If one of the factor of x² + x – 20 is (x + 5). Find the other
  15. If x + 2 is a factor of x³ – 2ax² + 16, then value of a is
  16. Using a suitable identity, determine the value of (17)³ + (-12)³ + (-5)³
  17. Find the product: (x – 3y) (x + 3y) (x² + 9y²)
  18. Factorise: (a – b)³ + (b – c)³ + (c – a)³
  19. If x + y = 12 and xy = 32, Find the value of x² + y²
  20. Find the value of x³ + y³ + z³ – 3xyz if x² + y² + z² = 83 and x + y + z = 15
  21. Find the value of 9x² + 4y² if xy = 6 and 3x + 2y = 12.
  22. Find a quadratic polynomial, the sum and product of whose zeroes are √2 and -3/2, respectively. Also find its zeroes.
  23. α and β are zeroes of the quadratic polynomial x² – 6x + y. Find the value of ‘y’ if 3α + 2β = 20
  24. Find the quadratic polynomial if its zeroes are 0 and √5.
  25. Find the zeroes of the polynomial 4x^2 – 4x – 8. Also, establish a relationship between the zeroes and coefficients.
  26. Find the value of "p" from the polynomial x² + 3x + p, if one of the zeroes of the polynomial is 2.
  27. If the product of zeroes of the polynomial p(x)=3x² + kx − 2 is 2/3 ​, find the value of k.
  28. If the zeroes of the quadratic polynomial p(x) = ax² + bx + c are reciprocal of each other, prove that c = a.
  29. Find a quadratic polynomial whose zeroes are 5 and −3.
  30. Find the zeroes of the polynomial: p(x) = x² − 7x + 10 and verify the relation between zeroes and coefficients.
  31. If one zero of the polynomial (a² + 9)x² + 13x + 6a is the reciprocal of the other, find the value of a.