Overview
This lecture covers factoring techniques, focusing on the difference of squares, factoring trinomials (including the AC method), conjugates, and simplifying complex fractions.
Difference of Two Squares
- The form ( a^2 - b^2 ) factors as ( (a-b)(a+b) ).
- Example: ( x^2 - 25 = (x-5)(x+5) ).
- Identify terms as perfect squares before factoring.
- With coefficients: ( 16 - 9y^2 = (4 - 3y)(4 + 3y) ).
Factoring Trinomials
- For ( x^2 + bx + c ), find two numbers that multiply to ( c ) and add to ( b ).
- Example: ( x^2 - 8x + 7 = (x-7)(x-1) ).
- For ( ax^2 + bx + c ), use the AC method: multiply ( a ) and ( c ), then factor by grouping.
- Example: ( 2x^2 + 11x + 5 ) — split ( 11x ) into ( 10x + x ), group, and factor to get ( (2x+1)(x+5) ).
Conjugates and the Difference of Squares
- Conjugates are pairs like ( a-b ) and ( a+b ).
- Multiplying conjugates gives a difference of squares: ( (a-b)(a+b) = a^2 - b^2 ).
- Example: ( (x-6)(x+6) = x^2 - 36 ).
- For roots: ( (4+\sqrt{y})(4-\sqrt{y}) = 16-y ).
Simplifying Complex Fractions
- Multiply numerator and denominator by the least common denominator to eliminate inner denominators.
- Cancel common factors across numerators and denominators.
- Always determine excluded values that make any denominator zero.
Key Terms & Definitions
- Difference of Squares — An expression of the form ( a^2 - b^2 ) factored as ( (a-b)(a+b) ).
- Trinomial — A polynomial with three terms, often written as ( ax^2 + bx + c ).
- AC Method — Multiplying the leading coefficient ( a ) and the constant ( c ) to aid in factoring.
- Conjugates — Expressions differing only by the sign between two terms (e.g., ( a+b ) and ( a-b )).
- Complex Fraction — A fraction where the numerator or denominator (or both) contain fractions.
Action Items / Next Steps
- Review assigned videos on factoring, conjugates, and complex fractions for additional practice.
- Prepare to factor various expressions and simplify complex fractions for the next lecture.