1. Introduction to Polynomials

A polynomial is an algebraic expression that consists of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication.

General Form of a Polynomial

A polynomial in one variable x is written as:


P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0
  • are constants (coefficients).
  • is the variable.
  • is a non-negative integer (degree of the polynomial).

Types of Polynomials

Polynomials can be classified based on:

(i) Degree of the Polynomial

The degree is the highest power of the variable in the polynomial.

  • Constant Polynomial: Degree = 0 (e.g., )
  • Linear Polynomial: Degree = 1 (e.g., )
  • Quadratic Polynomial: Degree = 2 (e.g., )
  • Cubic Polynomial: Degree = 3 (e.g., )

(ii) Number of Terms

  • Monomial: Contains one term (e.g., )
  • Binomial: Contains two terms (e.g., )
  • Trinomial: Contains three terms (e.g., )

2. Zeros of a Polynomial

The zero(s) or root(s) of a polynomial are the values of for which .

Finding Zeros of a Polynomial

For a polynomial , solve the equation .

  • Example: Find the zeros of .
    • Factorize: .
    • Set each factor to zero: or .
    • Zeros are and .

3. Relationship Between Coefficients and Zeros

For a quadratic polynomial , let its zeros be and . Then:

  1. Sum of Zeros: .
  2. Product of Zeros: .

Example: For ,

  • ,
  • ,
    which matches the zeros found earlier.

For a cubic polynomial with zeros :

  1. .
  2. .
  3. .

4. Division Algorithm for Polynomials

If and are polynomials, where , then


P(x) = G(x) \times Q(x) + R(x)
  • = Dividend
  • = Divisor
  • = Quotient
  • = Remainder

Example: Divide by

  1. Divide by → Quotient: .
  2. Multiply: .
  3. Subtract: .
  4. Divide by → Quotient: .
  5. Multiply: .
  6. Subtract: .
  7. Divide by → Quotient: .
  8. Multiply: .
  9. Subtract: .

Thus,


Q(x) = x^2 - 2x + 3, \quad R(x) = 0.

5. Factorization of Polynomials

Factorization involves expressing a polynomial as a product of its factors.

Methods of Factorization

  1. By Taking Common Factor
    Example: .

  2. By Splitting the Middle Term
    Example: .

  3. By Using Identities

    • .
    • .
    • .

  4. By Using Remainder and Factor Theorem

    • If , then is a factor.

6. Important Theorems

(i) Remainder Theorem

If a polynomial is divided by , then the remainder is .

Example: Find remainder when is divided by .

  • .
  • So, remainder is 0, meaning is a factor.

(ii) Factor Theorem

If , then is a factor of .

Example: Check if is a factor of .

  • .
  • Since remainder is 0, is a factor.

These notes cover all key concepts in Polynomials for Class 10. Let me know if you need further explanations!