Introductory Algebra

1.1 – Simplifying Algebraic Expressions

Algebraic Expressions

We begin our exploration of Algebra with Algebraic Expressions. These are the ways that we mathematically describe something, whether we are dealing with a complex mathematical model, or a simple addition of two items, we describe the situation with expressions. Some expressions can be simplified from the way that they are presented to us. This is the idea of simplifying an algebraic expression. We will start talking about simplifying algebraic expressions with a discussion of Terms.


Terms

A Term is a number, or the product of numbers and variables. Examples of Terms are:

3 \hspace{1.5cm}4x \hspace{1.5cm}17xy \hspace{1.5cm}12x^2y^4 \hspace{1.5cm}-\frac{2x}{5}

 

Each of these terms has two parts. The first is the numerical coefficient, often referred to simply as the coefficient. This is the number that is multiplied as part of the term. The numerical coefficient of the term 15x^2y is 15.

The numerical coefficient of the term xyz is 1. (If there is not a number written, it is assumed to be the number 1)

The numerical coefficient of the term \dfrac{2x^3}{5} is \dfrac{2}{5}.

The second part of each term is the variable(s). Some terms may not have variables written. A variable is not a necessary part of a term. The variables are typically written with letters, to indicate a number that is either unknown, or a number that might change.

Find the coefficient and variables of the term 5xyz
The coefficient of this term is 5. There are three variables in this term, x, y, and z.

Find the coefficient and variables of the term ab^2r
The coefficient of the term is 1, and the variables are a, b, and r

Two or more terms are called Like Terms if they have the same variable(s) with the same exponents. The variables need not be in the same order, but the exponents have to match exactly. Examples of like terms are:

4x \text{ and } 5x, \hspace{2cm} 32y \text{ and } \frac{4}{7}y, \hspace{2cm} 3x^2y \text{ and } 5x^2y

Unlike terms are terms that don’t have the same variables with the same exponents.

Are the terms 3x^2y and 4xy^2 like terms?
No, the exponents do not exactly match.

Are the terms 4xy^2z and \frac{2}{3}zxy^2 like terms?
Yes! The variables are written in a different order, but they are the same, and their exponents match.

Combining Like Terms

When we have like terms in an algebraic expression, we can often combine them together. We can do this because of the distributive property.

THE DISTRIBUTIVE PROPERTY
a(b+c)=ab+ac
Typically, we see this property as it is written above, and use it from left to right. But an equation can also be used from right to left. Doing this will allow us to see why like terms can be combined.

Combine like terms: 3x+7x
Applying the distributive property from right to left, we can see that 3x+7x would be the result of having distributed an x through the sum of 3 and 7. So we can write it as the original sum:

x(3+7)

But we can see that we can add 3 and 7 together:

x(10)

And we can write the result with the numerical coefficient in front as

10x

We, of course, don’t have to write this process out with so many steps every time we do it. We can simply know that the distributive property is at play here, and quickly combine like terms in our head (as long as the coefficients are not horrible).

Combine like terms: 6xy^2+8xy^2
Because the terms are like, we can simply add, as we did previously, the coefficients together. We combine to have 14xy^2.

All terms in an expression need not necessarily be like terms. If this is the case, we will simply combine those that are like.

Combine like terms: 3x^2 + 5x + 7x^2
We see that there are two x^2 terms, so we combine them. We do not combine them with the x term! The simplified expression is: 10x^2+5x

Combine like terms: 4x^2+6x+5x^2+18x
We combine the x^2 terms together, and can also combine the x terms together to obtain: 9x^2+24x

Using the distributive property as written should also not be forgotten. It can be used to remove parentheses to free terms.

Combine like terms: 2(x+2)+3(x-4)
Here, our like terms are stuck in separate parentheses. We can free them by applying the distributive property first:

2(x+2)+3(x-4)

2x+4+3x-12

Our terms are now free to combine. The like terms are not next to each other, but due to the commutative property of addition, we can add in whatever order we like. Our result:

5x-8

Apply the distributive property and then combine like terms  -(4x+2y-1)+2(5y-2x+7)
Note that the negative in front of the first parentheses must be distributed through. We can think of this as distributing a negative 1 throughout the first parentheses.

 -(4x+2y-1)+2(5y-2x+7)

-4x-2y+1 + 10y-4x+14

-8x+8y+15

Writing English Phrases as Algebraic Expressions

The biggest goal of math courses like the one you are in are to bring some sort of mathematical ability into the real world. Part of being able to use math in the real world requires you to be able to easily move between English (at least, if you are reading this book) and Mathematical notation.

Write “the sum of a number and five” in mathematical notation
We have a sum, and one of the numbers is unknown. When this is the case, we can input a variable for the unknown. Our mathematical expression is then x+5

Write “Three less than an unknown” in mathematical notation
This is written as x-3 Notice that this is three less than the unknown, and “3-x” is \textbf{\color{red}NOT} the same thing!

Write “four more than the product of three and an unknown” in mathematical notation
3x+4

Write as a mathematical expression, then simplify the result: “six added to the product of four and the sum of a number and two”
6+4(x+2)

6+4x+8

14+4x