Special Product Notes: Square of a Binomial & Product of Sum and Difference

LEARNING OBJECTIVES

  • I can identify patterns in special products.
  • I can apply special product formulas in multiplying polynomials.
  • I can simplify expressions using special products.

SPECIAL PRODUCT OVERVIEW

  • Special products are special forms or patterns of algebraic expressions. Recognizing these patterns helps students multiply faster without having to use the distributive property or FOIL method.

SQUARE OF A BINOMIAL

  • The square of a binomial is the product of a binomial multiplied by itself. In simple terms, squaring a two-term expression.
  • Formula: (a + b)^2
  • Shortcut steps:
    • Square the first term: a^2
    • Double the product of the first and last term: 2ab
    • Square the last term: b^2
    • Therefore, (a + b)^2 = a^2 + 2ab + b^2
  • Note: This expansion works for any a, b; when b is negative, the same formula expands to the correct result (e.g., $(x-6)^2 = x^2 - 12x + 36$).

PRACTICE: SQUARE OF A BINOMIAL

  • 1) (x + 5)^2 = x^2 + 10x + 25
  • 2) (x + 12)^2 = x^2 + 24x + 144
  • 3) (x - 6)^2 = x^2 - 12x + 36
  • 4) (x - 4)^2 = x^2 - 8x + 16
  • 5) (x + 10)^2 = x^2 + 20x + 100

PRODUCT OF SUM AND DIFFERENCE

  • Here’s a shortcut on the product of sum and difference: the operation between the first and last terms is subtraction, and the product equals the difference of their squares.
  • Explanation: The product of the sum and difference of the same two terms means multiplying one binomial with a plus sign and another with a minus sign using the same terms.
  • Formula: (a + b)(a - b) = a^2 - b^2
  • Shortcut: Square the first term, square the last term, then subtract: a^2 - b^2
  • Note: The operation between the first and last terms is always subtraction.

PRACTICE: PRODUCT OF SUM AND DIFFERENCE

  • 1) (2 + 6)(2 - 6)
  • 2) (2 + 4)(2 - 4)
  • 3) (2 + 5)(2 - 5)
  • 4) (-11 + 11)(-11 - 11)
  • 5) (-8 + 8)(-8 - 8)
  • Solutions:
    • (2 + 6)(2 - 6) = 2^2 - 6^2 = 4 - 36 = -32
    • (2 + 4)(2 - 4) = 2^2 - 4^2 = 4 - 16 = -12
    • (2 + 5)(2 - 5) = 2^2 - 5^2 = 4 - 25 = -21
    • (-11 + 11)(-11 - 11) = 0
    • (-8 + 8)(-8 - 8) = 0

EXTRA NOTES

  • These special products speed up multiplication by avoiding the distributive property or FOIL for two-term expressions.
  • The patterns connect to the foundational idea of factoring and the difference of squares.

CLOSING REMARKS

  • THANK YOU FOR LISTENING!!!