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(a + b)^2
  • Shortcut steps:
    • Square the first term: a2a^2
    • Double the product of the first and last term: 2ab2ab
    • Square the last term: b2b^2
    • Therefore, (a+b)2=a2+2ab+b2(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=x2+10x+25(x + 5)^2 = x^2 + 10x + 25
  • 2) (x+12)2=x2+24x+144(x + 12)^2 = x^2 + 24x + 144
  • 3) (x−6)2=x2−12x+36(x - 6)^2 = x^2 - 12x + 36
  • 4) (x−4)2=x2−8x+16(x - 4)^2 = x^2 - 8x + 16
  • 5) (x+10)2=x2+20x+100(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)=a2−b2(a + b)(a - b) = a^2 - b^2
  • Shortcut: Square the first term, square the last term, then subtract: a2−b2a^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 + 6)(2 - 6)
  • 2) (2+4)(2−4)(2 + 4)(2 - 4)
  • 3) (2+5)(2−5)(2 + 5)(2 - 5)
  • 4) (−11+11)(−11−11)(-11 + 11)(-11 - 11)
  • 5) (−8+8)(−8−8)(-8 + 8)(-8 - 8)
  • Solutions:
    • (2 + 6)(2 - 6) = 22−62=4−36=−322^2 - 6^2 = 4 - 36 = -32
    • (2 + 4)(2 - 4) = 22−42=4−16=−122^2 - 4^2 = 4 - 16 = -12
    • (2 + 5)(2 - 5) = 22−52=4−25=−212^2 - 5^2 = 4 - 25 = -21
    • (-11 + 11)(-11 - 11) = 00
    • (-8 + 8)(-8 - 8) = 00

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!!!