Overview
This lecture focused on factoring trinomials using the Diamond Method, especially when the leading coefficient (a) is 1, with step-by-step strategies, special cases, and common pitfalls.
Recognizing Trinomials to Factor
- Identify polynomials with three terms and no greatest common factor (GCF) other than 1.
- Ensure the polynomial is in order: highest power first, then variable, then constant.
- Check that the first term is positive; factor out a negative if not.
The Diamond Method (Factoring Trinomials)
- Use the Diamond Method for trinomials of the form ax² + bx + c.
- Write b (middle coefficient) on top and a × c (product of first and last coefficients) on the bottom.
- Find two numbers that add to b and multiply to a × c.
- If a = 1, directly write factors as (x + number1)(x + number2).
- For variables other than x, use the correct variable in factors.
Special Cases and Tips
- For trinomials with variables other than x or with multiple variables, apply the method, keeping variable consistency.
- If you cannot find two numbers meeting the add/multiply condition, the trinomial is called "prime" and cannot be factored.
- Always check for and factor out the GCF before applying the Diamond Method.
- For polynomials with higher powers or mixed variables, ensure the method applies, sometimes adjusting the order.
Factoring with GCF and Further Steps
- Always factor out the GCF before using the Diamond Method.
- After factoring out the GCF, reapply the method to the remaining trinomial.
- Combine all factored parts for the complete factorization.
Key Terms & Definitions
- Trinomial — a polynomial with exactly three terms.
- Greatest Common Factor (GCF) — the largest factor shared by all terms in a polynomial.
- Diamond Method — a graphical organizer to find two numbers that add to b and multiply to a × c in a trinomial.
- Prime Trinomial — a trinomial that cannot be factored further over the integers.
Action Items / Next Steps
- Practice factoring provided trinomials using the Diamond Method.
- Always check for GCF and correct order before starting.
- Try factoring both standard and special case trinomials for proficiency.