Use the Language of Algebra

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 a counting number and 2.

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 3 into 15, we get a counting number. In fact, $15÷3$ is 5, so 15 is $5·3.$

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 $8·9=72,$ we say that 8 and 9 are factors of 72. When we write $72=8·9,$ we say we have factored 72.

Other ways to factor 72 are $1·72,2·36,3·24,4·18,$ and $6·12.$ The number 72 has many factors: $1,2,3,4,6,8,9,12,18,24,36,$ and $72.$

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.

Example 1.2

How to Find the Prime Factorization of a Composite Number

Factor 48.

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Solution

We say $2·2·2·2·3$ is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer.

If we first factored 48 in a different way, for example as $6·8,$ the result would still be the same. Finish the prime factorization and verify this for yourself.

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.

Example 1.3

How to Find the Least Common Multiple Using the Prime Factors Method

Find the least common multiple (LCM) of 12 and 18 using the prime factors method.

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Solution

Notice that the prime factors of 12 $(2·2·3)$ and the prime factors of 18 $(2·3·3)$ are included in the LCM $(2·2·3·3).$ 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 $x,y,a,b,c.$

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 “>”.

Inequality

The expressions $a<b$ or $a>b$ 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 $2·2·2·2·2·2·2·2·2.$ This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write $2·2·2$ as $2^3$ and $2·2·2·2·2·2·2·2·2$ as $2^9.$ In expressions such as $2^3,$ 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.

Exponential Notation

We say $2^3$ is in exponential notation and $2·2·2$ is in expanded notation.

$a^n$ means multiply a by itself, n times.

The expression $a^n$ is read a to the $n^{th}$ power.

While we read $a^n$ as $“a$ to the $n^{th}$ power”, we usually read:

We’ll see later why $a^2$ and $a^3$ have special names.

Table 1.1 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 $4·2+1$ we would first multiply $4·2$ 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 $4+3·7.$ Some students simplify this getting 49, by adding $4+3$ and then multiplying that result by 7. Others get 25, by multiplying $3·7$ 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.

Example 1.4

Simplify: $18÷6+4\left(5-2\right).$

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Solution

Parentheses? Yes, subtract first.
Exponents? No.
Multiplication or division? Yes.
Divide first because we multiply and divide left to right.
Any other multiplication or division? Yes.
Multiply.
Any other multiplication of division? No.
Any addition or subtraction? Yes.
Add.

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

Example 1.5

Simplify: $5+2^3+3\left[6-3\left(4-2\right)\right].$

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Solution

Are there any parentheses (or other grouping symbols)? Yes.
Focus on the parentheses that are inside the brackets. Subtract.
Continue inside the brackets and multiply.
Continue inside the brackets and subtract.
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes. Simplify exponents.
Is there any multiplication or division? Yes.
Multiply.
Is there any addition of subtraction? Yes.
Add.
Add.

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.

Example 1.6

Evaluate when $x=4:$ ⓐ $x^2$ ⓑ $3^x$ ⓒ $2x^2+3x+8.$

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Solution

ⓐ

Use definition of exponent.
Simplify.

ⓑ

Use definition of exponent.
Simplify.

ⓒ

Follow the order of operations.

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 $7,y,5x^2,9a,$ and $b^5.$

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 $3x$ is 3. When we write $x,$ the coefficient is 1, since $x=1·x.$

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,

$7$ and $4$ are like terms.

$5x$ and $3x$ are like terms.

$n^2$ and $9n^2$ 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.

Example 1.7

How To Combine Like Terms

Simplify: $2x^2+3x+7+x^2+4x+5.$

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Solution

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. Table 1.2 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.

Example 1.8

Translate each English phrase into an algebraic expression:

ⓐ the difference of $14x$ and 9 ⓑ the quotient of $8y^2$ and 3 ⓒ twelve more than $y$ ⓓ seven less than $49x^2$

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Solution

ⓐ The key word is difference, which tells us the operation is subtraction. Look for the words of and and to find the numbers to subtract.

ⓑ The key word is quotient, which tells us the operation is division.

ⓒ The key words are more than. They tell us the operation is addition. More than means “added to.”

$$\begin{array}{c}\hfill \text{twelve more than}y\hfill \\ \hfill \text{twelve added to}y\hfill \\ \hfill y+12\hfill \end{array}$$

ⓓ The key words are less than. They tell us to subtract. Less than means “subtracted from.”

$$\begin{array}{c}\hfill \text{seven less than}49x^2\hfill \\ \hfill \text{seven subtracted from}49x^2\hfill \\ \hfill 49x^2-7\hfill \end{array}$$

We look carefully at the words to help us distinguish between multiplying a sum and adding a product.

Example 1.9

Translate the English phrase into an algebraic expression:

ⓐ eight times the sum of x and y ⓑ the sum of eight times x and y

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Solution

There are two operation words—times tells us to multiply and sum tells us to add.

ⓐ Because we are multiplying 8 times the sum, we need parentheses around the sum of x and y, $\left(x+y\right).$ This forces us to determine the sum first. (Remember the order of operations.)

$$\begin{array}{c}\hfill \text{eight times the sum of}x\text{and}y\hfill \\ \hfill 8(x+y)\hfill \end{array}$$

ⓑ To take a sum, we look for the words of and and to see what is being added. Here we are taking the sum of eight times x and y.

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.