Lecture Notes: Factoring Trinomials in Grade 8 Mathematics

Introduction

• Instructor: Joshua
• Focus: Sharpening skills in Grade 8 mathematics, particularly in factoring polynomials.
• Previous topic: Factoring special products of polynomials (Difference of two squares, Sum of two cubes, Difference of two cubes).
• Today's topic: Factoring trinomials.

Overview of Trinomials

• A trinomial consists of three terms.
• Perfect Square Trinomials:
  • Result from squaring binomials (e.g., ((a+b)^2 = a^2 + 2ab + b^2) and ((a-b)^2 = a^2 - 2ab + b^2)).
  • Identified as perfect square trinomials.

Example: Perfect Square Trinomial

• Polynomial: (N^2 + 16n + 64)
• Steps to factor:
  1. Check if first and last terms are perfect squares: (N^2) and (64 = 8^2)
  2. Check middle term: (2 \cdot N \cdot 8 = 16n)
  3. Conclusion: Polynomial is a perfect square trinomial; factored as ((N + 8)^2).

Example of Non-Perfect Square Trinomial

• Polynomial: (8x^2 - 24xy + 18y^2)
• Not a perfect square trinomial:
  • Check terms: (8x^2) and (18y^2) are not perfect squares.
  • Factor out greatest common monomial: (2(4x^2 - 12xy + 9y^2)).
  • Check if further factored: (2(2x - 3y)^2).

General Trinomials Factoring Steps

1. Identify terms:
   • Leading term, Middle term, Last term with signs.
2. Multiply leading and last terms: e.g., (x^2 \times 16 = 16x^2).
3. Find factors of the product whose sum equals the middle term.
4. Rewrite polynomial using identified factors.
5. Factor by grouping.

Practice Example: General Trinomials

• Polynomial: (x^2 + 10x + 16)
• Identify terms: positive leading term, middle term, and last term.
• Check if it’s a perfect square trinomial:
  • Not a perfect square (middle term not matching).
  • Factor as demonstrated through pairs and grouping.

Additional Example

• Polynomial: (2x^2 + 5x - 3)
• Identify terms: Leading term, Middle term, Last term.
• Multiply leading and last terms: (2x^2 \cdot -3 = -6x^2).
• Identify factors: Positive and negative combinations to achieve middle term.

Recap of Key Concepts

• Perfect Square Trinomials: Check for perfect square terms and validate against middle term.
• General Trinomials: Identify individual terms, determine products, factor using grouping.

Assessment Questions

1. Identify a perfect square trinomial.
2. Determine the factored form of a given trinomial.

Conclusion

• Importance of practice in mastering factoring techniques.
• Next lesson: Solving problems involving factoring polynomials.
• Encouragement to keep practicing using self-learning modules.