Introduction to Polynomial Factorisation

Introduction to Polynomial Factorisation

• Polynomial factorisation is the process of expressing a polynomial as a product of its factors.
• Factors can be numbers, variables, or other polynomials.

Types of Polynomials

1. Monomial: A single term (e.g. (3x^2)).
2. Binomial: Two terms (e.g. (x^2 + 5)).
3. Trinomial: Three terms (e.g. (x^2 + 5x + 6)).
4. Multinomial: More than three terms.

Basic Factorisation Techniques

1. Common Factor Extraction

   • Identify and pull out the greatest common factor (GCF).
   • Example: (6x^3 + 9x^2 = 3x^2(2x + 3)).

2. Difference of Squares

   • Use the identity: (a^2 - b^2 = (a + b)(a - b)).
   • Example: (x^2 - 9 = (x + 3)(x - 3)).

3. Perfect Squares

   • Recognize: (a^2 + 2ab + b^2 = (a + b)^2) or (a^2 - 2ab + b^2 = (a - b)^2).
   • Example: (x^2 + 6x + 9 = (x + 3)^2).

4. Trinomials

   • Factor using techniques like grouping or trial and error.
   • Example: (x^2 + 5x + 6 = (x + 2)(x + 3)).

5. Grouping

   • Group terms in pairs and factor out common factors.
   • Example: (ax + ay + bx + by = (a + b)(x + y)).

Special Polynomial Forms

1. Sum of Cubes

   • Identity: (a^3 + b^3 = (a + b)(a^2 - ab + b^2)).
   • Example: (x^3 + 8 = (x + 2)(x^2 - 2x + 4)).

2. Difference of Cubes

   • Identity: (a^3 - b^3 = (a - b)(a^2 + ab + b^2)).
   • Example: (x^3 - 27 = (x - 3)(x^2 + 3x + 9)).

Factorisation Techniques for Higher Degree Polynomials

• Polynomial long division: Used if a polynomial is divided by a binomial.
• Synthetic division: A shortcut for dividing polynomials.

Conclusion

• Factorisation is a crucial skill in algebra, useful for solving equations, simplifying expressions, and understanding polynomial behavior.
• Practice applying various techniques to develop proficiency in polynomial factorisation.

Polynomial Factorisation

• Polynomial factorization is the process of expressing a polynomial as a product of factors.
• Factors can be numbers, variables, or other polynomials.

Types of Polynomials

• Monomial: A single term (e.g., (3x^2)).
• Binomial: Two terms (e.g., (x^2 + 5)).
• Trinomial: Three terms (e.g., (x^2 + 5x + 6)).
• Multinomial: More than three terms.

Basic Factorisation Techniques

• Common Factor Extraction:
  • Identify and pull out the greatest common factor (GCF).
  • Example: (6x^3 + 9x^2 = 3x^2(2x + 3)).
• Difference of Squares:
  • Use the identity: (a^2 - b^2 = (a + b)(a - b)).
  • Example: (x^2 - 9 = (x + 3)(x - 3)).
• Perfect Squares:
  • Recognize: (a^2 + 2ab + b^2 = (a + b)^2) or (a^2 - 2ab + b^2 = (a - b)^2).
  • Example: (x^2 + 6x + 9 = (x + 3)^2).
• Trinomials:
  • Factor using techniques like grouping or trial and error.
  • Example: (x^2 + 5x + 6 = (x + 2)(x + 3)).
• Grouping:
  • Group terms in pairs and factor out common factors.
  • Example: (ax + ay + bx + by = (a + b)(x + y)).

Special Polynomial Forms

• Sum of Cubes:
  • Identity: (a^3 + b^3 = (a + b)(a^2 - ab + b^2)).
  • Example: (x^3 + 8 = (x + 2)(x^2 - 2x + 4)).
• Difference of Cubes:
  • Identity: (a^3 - b^3 = (a - b)(a^2 + ab + b^2)).
  • Example: (x^3 - 27 = (x - 3)(x^2 + 3x + 9)).

Factorisation Techniques for Higher Degree Polynomials

• Polynomial long division: Used if a polynomial is divided by a binomial.
• Synthetic division: A shortcut for dividing polynomials.

Conclusion

• Factorization is a crucial skill in algebra.
• It is useful for:
  • Solving equations
  • Simplifying expressions
  • Understanding polynomial behavior.
• Practice applying various techniques to develop proficiency in polynomial factorization.