Overview
This lecture explains how to identify and factor perfect square trinomials, with criteria, examples, and step-by-step factoring methods.
Identifying Perfect Square Trinomials
- The first and last terms must both be perfect squares and positive.
- The middle term must be twice the product of the square roots of the first and last terms.
- If the above conditions are met, the expression is a perfect square trinomial.
- Example: (x^2 + 2xy + y^2) is a perfect square trinomial; (x^2 + 5x + 6) is not.
Factoring Perfect Square Trinomials
- Given (x^2 + 2xy + y^2), the factored form is ((x + y)^2).
- If the middle term is negative as in (x^2 - 2xy + y^2), the factored form is ((x - y)^2).
- For trinomials like (4x^2 + 20x + 25), factor as ((2x + 5)^2).
- Always confirm both end terms are perfect squares and the middle term fits the twice-the-product rule.
Handling Non-Perfect Square Trinomials
- If the coefficients are not perfect squares, look for a greatest common factor (GCF) to factor out.
- After factoring out the GCF, check if the remaining trinomial is a perfect square trinomial.
- Example: (27a^2 + 72ab + 48b^2) factor out 3 to get (9a^2 + 24ab + 16b^2), which is a perfect square trinomial.
Step-by-Step Factoring Process
- Check if first and last terms are perfect squares and positive.
- Take square roots of first and last terms.
- Multiply these roots by 2; if it matches the middle term (with sign), it is a perfect square trinomial.
- Factored form: ((\text{square root of first term} \pm \text{square root of last term})^2).
Key Terms & Definitions
- Perfect Square Trinomial — A trinomial where the first and last terms are perfect squares, and the middle term is twice the product of their square roots.
- Factored Form — Expressing a trinomial as a binomial squared (((a \pm b)^2)).
- Greatest Common Factor (GCF) — The largest factor that can be evenly divided from all terms in an expression.
Action Items / Next Steps
- Practice identifying and factoring perfect square trinomials from homework problems.
- Review definitions and steps for checking if a trinomial is a perfect square.
- Attempt factoring non-perfect square trinomials by first factoring out the GCF.