TYP Math

Class Notes

• 1/16/06-1/20/06 Beyond PEMDAS - Algebraic Expressions, Precedence, Associativity, and Syntax Trees.
• 1/23/06-1/27/06 Simplification of Polyomials from first principles. We practiced simplifying multivariate expressions involving the addition and subtraction of signed monomials in several foundational steps:
  • replace all subtractions by additions of negated terms and transform the expression into a sum of positive and negative monomials (this is trickier than it looks when you consider that expressions are trees!)
  • use the associativity and commutativity of addition to reorder the terms so as to group like-terms together.
  • Use the Distributive Law (in reverse) to factor the variables out of the like terms, replacing each group of such terms by a parenthesized arithmetic expression multiplied by a product of variables raised to powers.
  • Evaluated the arithmetic expressions to get simple integers for the coefficients of the polynomial
  • Write the polynomial in standard form (replacing additions of negated terms with subtractions)
  We also worked on more direct methods in which you perform several of these steps "in your head," but making sure that we understood what was happening "under the hood"
• 1/30/06-2/3/06 Applications of the Distributivity of Multiplication over Addition. This week we expanded the class of expressions to include products of parenthesized expressions with monomials, e.g.

  a - b*(a-b*(a-b*(a-b)))
  

  and again looked at various approaches to simplifying these expressions, starting with the first principles approach and progressing to the "mental math" approach. One approach from basic principles is to use the transformation:

  X-Y*Z = X + (-Y)*Z
  

  to replace all of the subtractions with additions and hence lessen the chance of making sign errors:

  a - b*(a-b*(a-b*(a-b))) =
  a + (-b)*(a + (-b)*(a + (-b)*(a + (-b)))) =
  a + (-b)*(a + (-b)*(a + (-b)*a + (-b)*(-b)))) = 
  a + (-b)*(a + (-b)*(a + (-ab) + b^2)) =
  a + (-b)*(a + (-b)*a + (-b)*(-ab) + (-b)*b^2) =
  a + (-b)*(a + (-ab) + ab^2 + (-b^3)) =
  a + (-b)*a + (-b)*(-ab) + (-b)*ab^2 + (-b)*(-b^3) =
  a + (-ab) + ab^2 + (-ab^3) + b^4 =
  a - ab + ab^2 - ab^3 + b^4
  

  the more direct approach is to work with the signed expressions directly:

  a-b*(a-b*(a-b*(a-b))) =
  a-b*(a-b*(a-ab+b^2)) = 
  a-b*(a-ab+ab^2-b^3) =
  a-ab+ab^2-ab^3+b^4
  

  with the understanding that we are implicity converting the subtraction to additions of negated monomials "in our heads."
• 2/6/06-2/10/06