3.1 - Introduction to expressions



What is an expression?


A mathematical expression is a set of numbers and letters that represent numbers, combined using mathematical operations such as addition, subtraction, multiplication, division, roots, powers and functions.


Notice that an = sign is not allowed. Including one would create an equation.

The section “Introduction: Exactly what is algebra?” gave a very nice example of an expression. Here are some other examples of expressions:
The first expression could represent the total amount that a customer would pay if they were buying n items, each costing $15.00 plus an additional $4.00 for shipping and with a 5% tax on the entire amount. The second expression could represent the AC voltage delivered to your house by the electric utility. The last expression could represent the path followed by a thrown ball.



Evaluating an expression


To evaluate an expression means to assign values to all the letters that represent numbers and then carry out the mathematical operations (obeying the rules for the order of operations) to yield a numerical value for the expression.


For example if we let n equal 4 in the first expression above, 1.05 (15.00 n + 4.00), then the expression would become 1.05 (15.00 · 4 + 4.00), which simplifies to 67.20, meaning that the customer pays $67.20 for 4 items.


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The parts and properties of an expression

Just as a sentence can be broken down in various parts such as subject, verb and object, so a mathematical expression can be broken down into various parts. In this section we define and give examples of the following parts and properties of an expression:

Definition: Any letters in an expression that are allowed to change value during the course of a problem are called variables. Any letters that don't change value during a problem (but may change value from one problem to another problem) are called constants. Numbers, of course, are also constants.


Example: The expression ½ g t 2 describes the distance a falling object has fallen t seconds after it has been released. In a typical problem g is a constant with the value 9.81 and t is a variable to which we assign different values of time. Why don't we just always use the number 9.81 instead of the letter g? Because in a different problem we may want to study objects falling on the moon. The same equation applies; the only difference is that on the moon g has the value 1.6.



Definition: An expression can always be considered to be made up of parts that are added or subtracted together. Those parts are called terms.


Examples. Notice that the brackets in the second and third expressions cause the number of terms in those expressions to be 2 and 1.


Definition:
  • A monomial is an expression containing 1 term.
  • A binomial is an expression containing 2 terms.
  • A trinomial is an expression containing 3 terms.
  • A multinomial is any expression containing more than 1 term.
  • A polynomial is a special type of multinomial in which each term is a constant times a variable raised to a non-negative integer power. The variable must be the same for all the terms.

Examples:

Definition: A monomial or term can always be considered to be made up of parts that are multiplied together. Those parts are called the factors of the monomial. (Click here to review factors of a natural number, on which this is based.) Also, a monomial can always be considered to break into two factors: one factor containing only constants and one factor containing only variables. The constant factor is called the coefficient of the monomial. If there is no constant factor then the coefficient is understood to be 1.


Examples. In the following examples assume that a is a constant and x and y are variables.

Definition of degree:
  • The degree of a term or monomial is the power to which the variable is raised.
  • If the term or monomial contains several variables then the degree is the sum of the powers of all the variables.
  • The degree of a multinomial or polynomial is the degree of the term with the highest degree.

Examples: In the following examples assume that a and b are constants and x and y are variables.

Definition:   Like terms are terms that differ only in their coefficients; they have identical variable factors. In the next section, 3.2, we will see that like terms can be added by adding their coefficients.


Examples: Assume that x and y are variables. The like terms are shown in red.
The coefficients of the like terms are 4 and 9.
The coefficients of the like terms are 4 and  − 1.
If a and b are constants then the indicated terms ARE like terms and a and b are the coefficients. If a and b are variables then the indicated terms are NOT like terms.


Making a substitution and using an alias


Making a substitution means replacing a variable in an expression with a given expression.


This is similar to evaluating an expression except that you substitute in an expression rather than a number. It is usually necessary to put brackets around the substituted expression to preserve the proper order of operations.

Example 1: Substitute 2 x + y  for the variable t in the expression t 2 + 9.
(2 x + y) 2 + 9

Example 2: Typing
A = 3 x + y,     B = x − 2 y,     A B
into the Algebra Coach program and clicking on the Substitute button will substitute 3 x + y in for the variable A and x − 2 y in for the variable B in the expression A B. The result is:


In mathematics an alias is a single symbol (usually a capital letter like Q, short for the word Quantity) that represents a given expression. Using an alias means replacing the given expression by the alias wherever the given expression occurs inside another expression. An alias is usually used to make an expression look simpler and allow one not to be distracted by details.


Example: Consider the expression Let us use the alias Q = (a + b) 2. Then the expression becomes:

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The distributive law


Let a, b and c be any numbers. The distributive law states that:
    a (b + c) = a · b + a · c
            expanding →
          ← factoring
If we convert the expression on the left to the expression on the right then we say that we are distributing or expanding the expression. (Distributing because we are handing out or distributing a factor of a to each term in the brackets. Expanding because we are replacing an expression with one term by one with two terms.)

On the other hand, if we convert the expression on the right to the expression on the left then we say that we are factoring the expression (because the new expression is a product of factors).


We use the distributive law all the time in everyday life. For example when we add $9 and $8 to get $17 we are factoring using the distributive law:
$9 + $8 = $(9+8) = $17
(Not convinced? Try adding $9 and 8¢ and see if you get 17 of anything!)

As another example, suppose we multiply 12 times 32 using long multiplication:
This is just a tabular way of doing this:
12 · 32 = 12 · (30 + 2) = 12 · 30 + 12 · 2 = 360 + 24 = 384
We are expanding using the distributive law at the step with the red = sign.



The distributive law generalized


The basic distributive law,
    a (b + c) = a b + a c
can be generalized in many useful ways. We present four variations of the left side and show how the right side changes as a result.

1. The multinomial factor (b + c) on the left side can actually have any number of terms and the monomial factor a simply distributes over all of them. (Click here for the proof.) For example:
a (b + c + d + e) = a b + a c + a d + a e

2. The multinomial factor on the left side can have terms that are subtracted instead of added. The result is that on the right side the corresponding terms are also subtracted. (Click here for the proof.) For example:
a (b − c − d + e) = a b − a c − a d + a e

3. The monomial factor on the left side can be negative. The result is that on the right side the terms have their signs reversed. (Click here for the proof.) For example:
a (bcd + e) = − a b + a c + a d − a e
A special case is when the monomial factor on the left side is −1, in which case we don't have to write the “1” factor and can simply write:
− (bcd + e) = − b + c + d − e

4. The left side can be the product of two multinomials. Each term of the first multinomial distributes onto each term of the second multinomial. Probably the most important example is the product of 2 binomials:
(a + b) (c + d ) = a c + a d + b c + b d
(Click here for the proof.) Some people use the acronym FOIL (First, Outer, Inner, Last) to remember what terms on the left side are multiplied to give each of the four terms on the right side. F = first term from each factor, O = outer terms of the factors, I = inner terms of the factors, L = last term from each factor.


F O I L: This picture shows how each term of the first factor distributes onto each term of the second factor. Match an arc with a term of the same color.




Proof that:     a (b + c + d + e) = a b + a c + a d + a e
a (b + c + d + e) ←    start with the left side
= a (b + Q) use the alias   Q = c + d + e
= a b + a Q use the distributive law
= a b + a (c + d + e) substitute back for Q and repeat the above process until there is nothing left to distribute
a b + a c + a d + a e         the final result is the right side


Proof that:     a (b − c − d + e) = a b − a c − a d + a e
a (b − c − d + e) ←    start with the left side
= a (b + (−c) + (−d ) + e) change subtraction to addition of a negative
a b + a (−c) + a (−d ) + a e       use the generalized distributive law proved just previous
a b + (−a c) + (−a d ) + a e       use the fact that a positive times a negative is negative
= a b − a c − a d + a e change the addition of a negative back into subtraction; the final result is the right side


Proof that:     −a (bcd + e) = − a b + a c + a d − a e
a (bcd + e) ←    start with the left side
Q (bcd + e) use the alias   Q = −a
= Q b − Q c − Q d + Q e use the generalized distributive law proved just previous
= (−ab − (−ac − (−ad + (−ae     substitute back for Q
= (−a b) − (−a c) − (−a d) + (−a e) use the fact that a negative times a positive is negative
= − a b + a c + a d − a e use the fact that adding a negative is subtracting and subtracting a negative is adding; the final result is the right side


Proof that:     (a + b) (c + d ) = a c + a d + b c + b d
(a + b) (c + d ) ←    start with the left side
= Q (c + d) use the alias   Q = (a + b)
= Q c + Q d use the distributive law
= (a + b) c + (a + b) d substitute back for Q and distribute again
a c + b c + a d + b d         the result, after rearranging the terms, is the right side

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