Simplifying expressions

Basics of simplification

First we will establish the basics of simplifying expressions, then we will explore the process for more complex questions all the way up to polynomials.

Organizing terms

The first thing you should do as you begin to simplify an expression is identify the different terms you have. Remember, a term is a section of numbers or variables all touching each other within an equation. This step is important to recognizing which terms are negative as a result of subtraction signs.

Then, you will combine like terms. Like terms are terms that have the same exponent in the variable or can be manipulated to appear that way. Let’s do an example before moving on, but we will go into more detail later in the chapter.

Simplify the following expression:

$$6x+x-3x+2$$

Identify terms: $6x,x,-3x,2$.

Combine like terms: $6x$, $x$, and $-3x$ all share the same exponent of $x$, so we can combine them to be $6x+x-3x$, which is $4x$. We cannot combine the value with $2$ because it’s the exponent of $x$ is zero. You can imagine it looks like this: $2x^0$

Answer: $4x+2$

Fractions

The best way to simplify a fraction is to find terms in the numerator that match terms in the denominator. Doing this means we can “cross them out” because a number divided by itself is equal to $1$. In the expression $\frac{4x}{2x}$ the $x$ in the numerator and denominator may be crossed out or considered to be equal to $1$, and be simplified as follows: $\frac{4(1)}{2(1)}$, which is equal to $2$.

The golden rule in simplifying fractions (and don’t worry, we’ll remind you later on as well) is that in order to cross anything out in a fraction, the entire addition/subtraction expression must be simplified. If you have two terms in the numerator or denominator added together, they may not be crossed out one at a time with the denominator. The whole term must be simplified at once. See the example below for a visualization.

$$\frac{(2x+7)}{2x}$$

In this example you cannot cross out the $2x$ because it would not simplify the entire expression of the numerator.

Simplification situations

You will typically have four situations when it comes to fraction simplification:

1. Addition/subtraction in neither the numerator nor the denominator

2. Addition/subtraction in both the numerator and denominator

3. Addition/subtraction in the numerator

4. Addition/subtraction in the denominator

We have explored situation 1 above, so now we will discuss the methods to perform the other situations.

For situation 2, where addition/subtraction exists in both the numerator and the denominator, we may only cross out or simplify entire expressions:

$$\frac{(2x+7)}{(2x+7)}$$

This expression may be simplified because there is a similar expression in both the numerator and denominator; you can cross it out to equal $\frac{1}{1}$ or simply $1$.

For situation 3, where addition/subtraction exists only in the numerator, we can simplify the fraction by splitting it apart. Take each individual term and divide them all by the same denominator, creating as many fractions as there are terms in the numerator:

$$\frac{(2x+7)}{2x}$$

This can be expanded to:

$$\frac{2x}{2x}+\frac{7}{2x}$$

This is sometimes helpful because then we can cross terms out one by one. For instance, we can simplify the left term, $\frac{2x}{2x}$, by crossing out the common $2x$ in both the numerator and denominator: $\frac{1}{1}$, which is simply $1$. We are not able to simplify the term on the right, but we managed to simplify the other quite well, leaving us with a final simplified term of

$$1+\frac{7}{2x}$$

For situation 4, where addition/subtraction exists only in the denominator, we cannot split the fraction apart. This is a very important rule. Do not make this mistake! It is an easy one to forget.

Polynomials and factoring

Simplifying polynomial expressions

Polynomials are expressions with variables having different exponents. An example of a polynomial expression is $4x+x^2-3+x$. In order to make simplifying these expressions easier for us, we will add a step to our method for basic simplifying: organization. We will reorganize the terms in descending order, which means we will put the terms with the highest exponents first, and the lowest exponents last:

$$4x+x^2-3+x=x^2+4x+x-3$$

This reorganization of terms allows us to combine like terms more quickly and easily. After combining like terms, we get $x^2+5x-3$. Does this look familiar? It is a quadratic equation. Most quadratic equations are organized in descending order to express that they are already simplified, or to help make simplification easier for you. Try an example of simplifying a polynomial expression. Identify, organize, then combine.

Simplify the following expression:

$$4-x^3+3x^2+4x^2-x+12+3x^3-8$$

Factoring

Factoring will be explored more in depth in the chapter [Factorization of quadratics and cubics]. Here, we will try to understand what happens during factorization.

The purpose of factoring is to find something in common between all the terms of an expression and remove it from each term in order to make them appear more simple. However, we do not get rid of the removed commonality. Instead, we place it outside of parentheses to show that it could be distributed back into the expression to revert to the original expression. For instance,

$$2x+4$$

This is an expression that has a number in common in each term: $2$. We could divide the whole expression by $2$ to simplify it, then we would place it outside the expression’s parentheses:

$$2(x+2)$$

If you distribute the $2$ back into the expression within the parentheses you will end up with the same expression we started with:

$$2x+x$$

You can also factor variables. In this case, when you want to remove a whole term like $x$, you must replace it with $1$. After all, you are dividing $x$ by $x$, which equals $1$.

$$x(2+1)$$

We can further simplify this expression as $3x$.

This skill is very important for simplifying fractions. If you can factor out a number to get a similar expression on the top and bottom of the fraction, then you can cross them out! Review this example:

$$\frac{3}{(6x+12)}$$

What can you factor out of the expression in the denominator that would help simplify the fraction (i.e., cross out a term)?

Key points

Simplifying expressions. Identify terms, organize terms in descending order, and combine like terms.

Simplifying fractions. You can cross out entire expressions if they exist on both the top and bottom of the fraction. If addition/subtraction is only in the numerator, you can expand the fraction by splitting up the terms with the same denominator. If addition/subtraction is only in the denominator, you may not do this. Try to factor something out of the expression in the denominator to cross it out with the numerator.

Factoring. Divide an entire expression by one number or variable that all terms have in common. Then, place the commonality outside of parentheses to show it may be distributed back into the expression. Now you can use the factored number or the simplified expression to cross out terms in a fraction.