Overview
This lecture explains how to identify and factor perfect square trinomials, including step-by-step examples and strategies for handling related polynomial expressions.
Identifying Perfect Square Trinomials
- A perfect square trinomial has the form: first term + middle term + last term.
- The first term must be a positive perfect square.
- The last term must be a positive perfect square.
- The middle term must be twice the product of the square roots of the first and last terms.
Examples of Identification
- x² + 2xy + y² is a perfect square trinomial because its terms fit the criteria.
- 4x² + 20x + 25 is a perfect square trinomial.
- x² + 5x + 6 is not a perfect square trinomial since 6 is not a perfect square.
- 9x² + 30xy + 25y² is a perfect square trinomial.
- 4x² + 2xy + y² is not a perfect square trinomial (middle term is incorrect).
Factoring Perfect Square Trinomials
- The general form: x² + 2xy + y² = (x + y)²; x² - 2xy + y² = (x - y)².
- For x² + 10x + 25, the factored form is (x + 5)².
- For 16x² + 72x + 81, the factored form is (4x + 9)².
- For x² - 22x + 121, the factored form is (x - 11)².
- For 25m² - 20mn + 4n², the factored form is (5m - 2n)².
Handling Non-Perfect Square Trinomials
- If the expression is not a perfect square trinomial, factor out the greatest common factor (GCF) first.
- After factoring out the GCF, check if the resulting trinomial is a perfect square and factor if possible.
Key Terms & Definitions
- Perfect Square Trinomial — A trinomial where the first and last terms are positive perfect squares and the middle term is twice their product.
- Factoring — Rewriting an expression as a product of its factors.
- Greatest Common Factor (GCF) — The highest factor that divides all terms in the expression.
Action Items / Next Steps
- Practice identifying and factoring perfect square trinomials.
- Factor out the GCF first if the trinomial does not initially appear perfect.
- Review and memorize the structure of perfect square trinomials for quick recognition.