Polynomial Product Formulas

Overview

This lecture introduces notable products in polynomials, focusing on the square of a binomial and the sum by difference formulas, including their applications and geometric interpretations.

Square of a Binomial

• Notable products are shortcuts to perform common polynomial operations quickly, useful for factoring.
• The square of a binomial formula: (a + b)² = a² + 2ab + b².
• The three terms represent: square of the first term, double the product of both terms, and square of the second term.
• Applies to any monomials substituted for a and b.
• Example: (2x + 3y)² = 4x² + 12xy + 9y².
• If the binomial is (a - b), the formula is (a - b)² = a² - 2ab + b².
• Sign of the double product (±2ab) depends on whether both monomials have the same or opposite signs.
• Squaring a sum differs from summing the squares: (a + b)² ≠ a² + b².
• Geometric interpretation: the area of a square with side (a + b) decomposes into a², 2ab, and b².

Sum by Difference (Product of a Sum and Difference)

• The sum by difference formula: (a + b)(a - b) = a² - b².
• The result is the difference of the squares of the two terms.
• Example: (3x + 2y)(3x - 2y) = 9x² - 4y².
• The order inside parentheses does not matter (sum first or difference first).
• Applies if the terms are sums themselves, e.g., (x + y + 1)(x + y - 1) = (x + y)² - 1.
• If both parentheses reverse signs (e.g., (a - b)(b - a)), the result is - (b - a)², not the difference of squares.
• Geometric interpretation: the area of a rectangle with sides (a + b) and (a - b) equals a² - b².

Key Terms & Definitions

• Notable products — Common polynomial multiplication shortcuts useful for simplification and factoring.
• Binomial — An algebraic expression with two terms (e.g., a + b).
• Square of a binomial — The product of a binomial multiplied by itself, yielding three terms.
• Sum by difference — The product (a + b)(a - b), resulting in the difference of squares.
• Monomial — An algebraic expression with one term.

Action Items / Next Steps

• Practice applying the square of a binomial and sum by difference formulas with different monomials.
• Prepare for the next lecture on the cube of a binomial and the square of a trinomial.