# Foundations

## Use the Language of Algebra

In algebra, we use a letter of the alphabet to represent a number whose value may change or is unknown. Commonly used symbols are a, b, c, m, n, x, and y. Further discussion of constants and variables appears later in this section.

### Find Factors, Prime Factorizations, and Least Common Multiples

The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. A multiple of 2 can be written as the product of 2 and a counting number.
Similarly, a multiple of 3 would be the product of a counting number and 3.
We could find the multiples of any number by continuing this process.
Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 15 by 3, we get a counting number. In fact,  is 5, so 15 is 
If we were to look for patterns in the multiples of the numbers 2 through 9, we would discover the following divisibility tests:
In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.
Since  we say that 8 and 9 are factors of 72. When we write  we say we have factored 72.
Other ways to factor 72 are  and  The number 72 has many factors:  and 
Some numbers, such as 72, have many factors. Other numbers have only two factors. A prime number is a counting number greater than 1 whose only factors are 1 and itself.
The counting numbers from 2 to 20 are listed in the table with their factors. Make sure to agree with the “prime” or “composite” label for each!
The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.
A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful in many topics in this course.
To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime. Otherwise it is easy to lose track of the prime numbers.
If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.
One of the reasons we look at primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.
To find the least common multiple of two numbers we will use the Prime Factors Method. Let’s find the LCM of 12 and 18 using their prime factors.
Notice that the prime factors of 12  and the prime factors of 18  are included in the LCM  So 36 is the least common multiple of 12 and 18.
By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.

### Use Variables and Algebraic Symbols

In algebra, we use a letter of the alphabet to represent a number whose value may change. We call this a variable and letters commonly used for variables are 
A number whose value always remains the same is called a constant.
To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We’ll list the symbols used to indicate these operations below.
When two quantities have the same value, we say they are equal and connect them with an equal sign.
On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols “<” and “>”.
The expressions  or  can be read from left to right or right to left, though in English we usually read from left to right. In general,
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an expression, which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols now.
Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.
What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. A sentence has a subject and a verb. In algebra, we have expressions and equations.
Notice that the English phrases do not form a complete sentence because the phrase does not have a verb.
An equation is two expressions linked by an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.
Suppose we need to multiply 2 nine times. We could write this as  This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write  as  and  as  In expressions such as  the 2 is called the base and the 3 is called the exponent. The exponent tells us how many times we need to multiply the base.
While we read  as  to the  power”, we usually read:
We’ll see later why  and  have special names.
 shows how we read some expressions with exponents.

### Simplify Expressions Using the Order of Operations

To simplify an expression means to do all the math possible. For example, to simplify  we would first multiply  to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:
By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.
We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.
For example, consider the expression  Some students simplify this getting 49, by adding  and then multiplying that result by 7. Others get 25, by multiplying  first and then adding 4.
The same expression should give the same result. So mathematicians established some guidelines that are called the order of operations.
Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase “Please Excuse My Dear Aunt Sally”.
It’s good that “My Dear” goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, “Aunt Sally” goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

### Evaluate an Expression

In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.

### Identify and Combine Like Terms

Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.
Examples of terms are  and 
The constant that multiplies the variable is called the coefficient.
Think of the coefficient as the number in front of the variable. The coefficient of the term  is 3. When we write  the coefficient is 1, since 
Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are like terms.
Look at the following 6 terms. Which ones seem to have traits in common?
We say,
 and  are like terms.
 and  are like terms.
 and  are like terms.
If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.

### Translate an English Phrase to an Algebraic Expression

We listed many operation symbols that are used in algebra. Now, we will use them to translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that.  summarizes them.
Look closely at these phrases using the four operations:
Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.
We look carefully at the words to help us  distinguish between multiplying a sum and adding a product.
Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.
The expressions in the next example will be used in the typical coin mixture problems we will see soon.

### Key Concepts

1. Divisibility Tests
A number is divisible by:
  2 if the last digit is 0, 2, 4, 6, or 8.
  3 if the sum of the digits is divisible by 3.
  5 if the last digit is 5 or 0.
  6 if it is divisible by both 2 and 3.
  10 if it ends with 0.
2. How to find the prime factorization of a composite number.


3. How To Find the least common multiple using the prime factors method.


4. Equality Symbol
 is read “a is equal to b.”
The symbol “=” is called the equal sign.
5. Inequality


6. Inequality Symbols


7. Grouping Symbols

8. Exponential Notation
 means multiply a by itself, n times.
The expression  is read a to the  power.
9. Simplify an Expression
To simplify an expression, do all operations in the expression.
10. How to use the order of operations.


11. How to combine like terms.





### Practice Makes Perfect

Identify Multiples and Factors
In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.
Find Prime Factorizations and Least Common Multiples
In the following exercises, find the prime factorization.
In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify each expression.
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Simplify Expressions by Combining Like Terms
In the following exercises, simplify the following expressions by combining like terms.
Translate an English Phrase to an Algebraic Expression
In the following exercises, translate the phrases into algebraic expressions.

### Writing Exercises


### Self Check

ⓐ Use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.