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.

This quiz covers the fundamentals of polynomial factorisation, including the various types of polynomials such as monomials, binomials, and trinomials. You will also learn key factorisation techniques, including common factor extraction and the difference of squares. Test your understanding of these concepts through a series of questions.