Algebraic Expansion of Binomials

Algebraic Expansion of Binomials

  

Introduction to Binomial Expansion

  • Binomials are algebraic expressions containing two terms.
  • The standard form of a binomial is represented as (a + b) or (a - b).

Expansion of Binomials

  • The goal is to expand the product of two binomials using the distributive property or the FOIL method.
Example 1: Expanding (x + 6)(x - 1)
  1. Apply the distributive property (also known as the FOIL method)
       - F (First): Multiply the first terms of each binomial:
         - ximesx=x2x imes x = x^2
       - O (Outside): Multiply the outside terms:
         - ximes(−1)=−xx imes (-1) = -x
       - I (Inside): Multiply the inside terms:
         - 6imesx=6x6 imes x = 6x
       - L (Last): Multiply the last terms of each binomial:
         - 6imes(−1)=−66 imes (-1) = -6
  2. Combine all the products:
       - The expanded form is:
    x2−x+6x−6x^2 - x + 6x - 6
       - Combine like terms:
    x2+5x−6x^2 + 5x - 6
Example 2: Expanding (2x + 1)(x - 5)
  1. Apply the distributive property (FOIL method)
       - F (First): Multiply the first terms:
         - 2ximesx=2x22x imes x = 2x^2
       - O (Outside): Multiply the outside terms:
         - 2ximes(−5)=−10x2x imes (-5) = -10x
       - I (Inside): Multiply the inside terms:
         - 1imesx=x1 imes x = x
       - L (Last): Multiply the last terms:
         - 1imes(−5)=−51 imes (-5) = -5
  2. Combine all the products:
       - The expanded form is:
    2x2−10x+x−52x^2 - 10x + x - 5
       - Combine like terms:
    2x2−9x−52x^2 - 9x - 5

Summary

  • The expansion of the binomial expressions can be thoroughly achieved using the distributive property or the FOIL method which makes it easier to handle multiplication of polynomials efficiently.
  • Final results of the examples are:
       - For (x + 6)(x - 1): x2+5x−6x^2 + 5x - 6
       - For (2x + 1)(x - 5): 2x2−9x−52x^2 - 9x - 5