Math 80 Lecture - Chapter 5                                                       Merced College; Don Power   

 

VARIABLE EXPRESSIONS

 

5.1  Properties of Real Numbers

 

Properties:  [list in text]

     

Tasks: 

 

Given a statement (an "identity," a statement that is always true) be able to identify which property is illustrated

           

Be able to complete an equation based on a property

 

Simplify variable expressions:

 

Addition/subtraction of 2-3 terms, or multiplication of 2-3 factors, some of which are fractions, most of which contain variables, selected to illustrate the properties

 

Distributive Law

 

Simplify statements using the distributive law

Distribute a negative across a parenthesis containing additions and subtractions

 

5.2  Variable Expressions in Simplest Form

 

Simplest form involves

      1.  Clearing parentheses (applying the distributive law), and

      2.   Collecting "like terms."

 

Algebraic terms include

      Coefficient (number with its sign) - may be implied 1

      Variable or variables (letter(s))

      Exponent for each variable - may be implied 1

 

Terms are alike if

      The variables with their exponents are identical

 

We collect like terms by

      1.  Adding the coefficients

      2.  Copying the variables with their exponents (don't change the exponents)

 

      Ex:  −3x2y + 7x2y = 4x2y

 

 

5.3  Adding and Subtracting Polynomials

 

What are polynomials?  The terms (previous lesson) have exponents that are whole numbers (and no square roots of variables)

 

Special polynomials:

      Monomials

      Binomials

      Trinomials

      More terms:  polynomials in general

 

Set the problems up with the polynomials in parentheses

      Addition:  (    ) + (     )

      Subtractions:  (     ) − (     )

 

Clear the parentheses (with the distributive law) and collect like terms.

 

You may use either

      Horizontal format (the usual way), or

      Vertical format

 

5.4a  Multiplication of Polynomials

 

For this lesson, we need some useful rules of exponents:

      Multiplication rule:      xm times xn = x m+n

      Division rule:               xm divided by xn = x m−n

 

Rationale:  x5 times x3 = x*x*x*x*x  times x*x*x = x*x* ... *x with eight total factors = x8

And for x5 / x3, we get x*x*x*x*x  divided by x*x*x.  Three factors will reduce, leaving x*x = x2

 

So, to multiply two monomials, e.g. (−4v2wz3)(3vwz7)

      Multiply the signs:  neg times pos is neg

      Multiply the numbers [i.e. their absolute values]:  4*3 is 12

      Copy the first base:  v

      Calculate (add) the exponents of the first base:  2 + implied 1 is 3, so we get v3

      Repeat for the other variables:  w*w is w2,  z3 * z7 is z10

      Result:  −12v3w2z10.

 

The same steps apply if we have 3 or more monomials being multiplied.

 

5.5a  Multiplying a polynomial by a monomial

 

Technique:  Distributive Law, using the techniques in 5.4a

Ex:  −3xy2 ( 5x2y3 − 2xy + 7y4 + y − 1 )

      = (−3xy2)(5x2y3) + (−3xy2)(−2xy) + (−3xy2)(7y4) + (−3xy2)(y) + (−3xy2)(−1)     Don't write this step

      = −15x3y5 + 6x2y3 − 21xy6 − 3xy3 + 3xy2

Remember:

      Multiply the signs

      Multiply the numbers [i.e. their absolute values]

      Copy each base (the variables)

      For each base, add the exponents (recall that a letter be itself has an implied exponent of 1)

 

 

5.7  Verbal expressions and Variable Expressions

 

A.  Translate short phrases into symbols

B.  Translate longer statements, including statements requiring combinations of operations

            Suggestion:  start inside the innermost parenthetical expression (at the end of the word picture) and work your way to the outside

C.  Applications:  These applications define one variable in terms of another, in words, and then ask you to do the same thing in symbols

 

 

Return to:  Merced College; Don Power               Updated 10/19/06 by Don Power