Overview
This lecture focuses on two notable products in algebra: the square of a binomial and the sum by difference, explaining their patterns, uses, and providing calculation examples.
Square of a Binomial
- The square of a binomial ((a + b)^2) equals (a^2 + 2ab + b^2).
- This expands as: (square of the first term) plus (double the product of both terms) plus (square of the second term).
- This rule applies for all monomials substituted as (a) and (b).
- If both terms have the same sign, the middle term ((2ab)) is positive; if signs differ, it's negative.
- Example: ((2x + 3y)^2 = 4x^2 + 12xy + 9y^2).
- Example: ((2x - 3y)^2 = 4x^2 - 12xy + 9y^2).
- Example: ((-2x^2 + \frac{1}{2}xy)^2 = 4x^4 - 2x^3y + \frac{1}{4}x^2y^2).
- Squaring a sum is not the same as summing the squares; the difference is the double product term.
- Geometric interpretation: The area of a square with side (a + b) is (a^2 + 2ab + b^2).
Sum by Difference (Product of Sum and Difference)
- The product ((a + b)(a - b) = a^2 - b^2).
- This equals (square of the first term) minus (square of the second term).
- Example: ((3x + 2y)(3x - 2y) = 9x^2 - 4y^2).
- Example: ((\frac{2}{3}x - 1)(\frac{2}{3}x + 1) = \frac{4}{9}x^2 - 1).
- Example: ((-x^3 + z)(-x^3 - z) = x^6 - z^2).
- Works even when terms in parentheses are expressions, e.g.: ((x+y+1)(x+y-1) = (x+y)^2 - 1).
- If both parentheses switch sign (((a-b)(b-a))), result is (-(b-a)^2), not a difference of squares.
- Geometric interpretation: The area of rectangle sides (a+b) and (a-b) equals (a^2 - b^2).
Key Terms & Definitions
- Binomial — An algebraic expression with two terms (e.g., (a + b)).
- Notable Products — Special algebraic expansions known for their patterns, used to simplify or factor polynomials.
- Double Product — The term (2ab), double the product of two terms in a binomial square.
- Sum by Difference — Product of the sum and difference of two terms: ((a+b)(a-b)).
Action Items / Next Steps
- Practice expanding and simplifying using these patterns for binomial squares and sum by difference.
- Prepare for next lecture on the cube of the binomial and the square of the trinomial.