Overview
This lecture explains how to identify and factor perfect square trinomials, including the necessary conditions and step-by-step factoring procedure.
Identifying Perfect Square Trinomials
- The first term must be a positive perfect square (e.g., (x^2), (4x^2), (9x^2)).
- The last (third) term must also be a positive perfect square (e.g., (y^2), (25), (25y^2)).
- The middle term must equal twice the product of the square roots of the first and last terms.
- If any term does not meet these criteria, the expression is not a perfect square trinomial.
Examples of Identification
- (x^2 + 2xy + y^2): Perfect square trinomial (middle term is (2xy)).
- (4x^2 + 20x + 25): Perfect square trinomial (middle term is (2 \times 2x \times 5 = 20x)).
- (x^2 + 5x + 6): Not a perfect square trinomial (6 is not a perfect square).
- (9x^2 + 30xy + 25y^2): Perfect square trinomial (middle term is (2 \times 3x \times 5y = 30xy)).
- (4x^2 + 2xy + y^2): Not a perfect square trinomial (middle term should be (4xy), not (2xy)).
Factoring Perfect Square Trinomials
- For (x^2 + 2xy + y^2), factored form is ((x + y)^2).
- For (x^2 - 2xy + y^2), factored form is ((x - y)^2).
- Sign of the middle term determines if the factor is (+) or (-).
- Examples:
- (x^2 + 10x + 25 \rightarrow (x + 5)^2)
- (16x^2 + 72x + 81 \rightarrow (4x + 9)^2)
- (x^2 - 22x + 121 \rightarrow (x - 11)^2)
- (25m^2 - 20mn + 4n^2 \rightarrow (5m - 2n)^2)
Factoring When Not Immediately a Perfect Square Trinomial
- If not all terms are perfect squares, factor out the greatest common factor (GCF).
- After factoring out GCF, check if the remaining trinomial is a perfect square.
- Example: (27a^2 + 72ab + 48b^2) factor out 3: (3(9a^2 + 24ab + 16b^2)), then factor to (3(3a + 4b)^2).
- Example: (36x^3 - 24x^2 + 36x), factor out (4x): (4x(x^2 - 6x + 9)), then factor to (4x(x - 3)^2).
Key Terms & Definitions
- Perfect Square Trinomial — A trinomial with form (a^2 + 2ab + b^2) or (a^2 - 2ab + b^2).
- Greatest Common Factor (GCF) — The largest expression that divides all terms in a polynomial.
Action Items / Next Steps
- Practice identifying and factoring perfect square trinomials from given expressions.
- Review classwork or textbook exercises on factoring special polynomials.