Overview
This lecture explains how to recognize and factor perfect square trinomials, using quick checks involving square roots and the middle term.
Recognizing Perfect Square Trinomials
- A perfect square trinomial has the form ( a^2 + 2ab + b^2 ) or ( a^2 - 2ab + b^2 ).
- To check if a trinomial is a perfect square, take the square roots of the first and last terms.
- The middle term must be ( 2 \times ) (square root of first term) ( \times ) (square root of last term).
Factoring Steps
- Identify if both first and last terms are perfect squares.
- Calculate the middle term using ( 2ab ) or ( -2ab ), matching the sign in the problem.
- If the calculation matches the actual middle term, the trinomial is a perfect square.
- Write the factor as ( (a \pm b)^2 ), where ( a ) and ( b ) are the square roots of the first and last terms, and the sign matches the middle term.
Checking Your Answer
- Always verify by expanding (FOIL) your factored expression to ensure it matches the original trinomial.
Key Terms & Definitions
- Perfect Square Trinomial — A quadratic of the form ( a^2 \pm 2ab + b^2 ) that factors into ( (a \pm b)^2 ).
- FOIL — Method to expand two binomials: First, Outer, Inner, Last.
Action Items / Next Steps
- Practice factoring trinomials by identifying perfect squares and checking with the middle term formula.
- Always expand factored forms to verify correctness.