Simplifying expressions

Basics of simplification

In this section, you’ll learn the core skills for simplifying expressions. We’ll start with basic expressions, then build up to fractions and polynomials.

Organizing terms

When you simplify an expression, start by identifying the terms. A term is a group of numbers and/or variables that are multiplied together (they’re “touching”), and terms are separated by addition or subtraction signs. Identifying terms also helps you keep track of which terms are negative because of subtraction.

Next, you combine like terms. Like terms have the same variable part (same variables raised to the same exponents). We’ll do a quick example here and return to this idea in more detail later.

Simplify the following expression:

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

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

Combine like terms: $6x$, $x$, and $-3x$ are all $x^1$ terms, so you can combine them:

• $6x+x-3x=4x$

You can’t combine $2$ with the $x$ terms because $2$ has no $x$ factor. (You can think of it as $2x^0$.)

Answer: $4x+2$

Fractions

A common way to simplify a fraction is to look for factors in the numerator that match factors in the denominator. When the same nonzero factor appears in both places, you can cancel it because a number divided by itself equals $1$.

For example, in $\frac{4x}{2x}$, the $x$ cancels:

• $\frac{4x}{2x}=\frac{4(1)}{2(1)}=2$

The key rule for simplifying fractions is this:

• You may only cancel factors, not individual pieces of a sum or difference.
• If the numerator or denominator contains addition/subtraction, you can’t cancel just one term from it.

See the example below:

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

In this example, you cannot cross out the $2x$ from the numerator, because $2x+7$ is a sum. The $2x$ is not a factor of the entire 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 explored situation 1 above, so now we’ll discuss methods for the other situations.

For situation 2, where addition/subtraction exists in both the numerator and the denominator, you may only cancel entire expressions:

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

This expression simplifies because the entire numerator matches the entire denominator. You can cancel to get $\frac{1}{1}$, which is $1$.

For situation 3, where addition/subtraction exists only in the numerator, you can sometimes simplify by splitting the numerator across the denominator. Divide each term in the numerator by the same denominator:

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

This can be expanded to:

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

Now you can simplify each fraction separately. The first fraction simplifies to $1$, and the second does not simplify further:

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

For situation 4, where addition/subtraction exists only in the denominator, you cannot split the fraction apart. This is an important rule:

• You may not rewrite $\frac{a}{b+c}$ as $\frac{a}{b}+\frac{a}{c}$.

Polynomials and factoring

Simplifying polynomial expressions

A polynomial is an expression with terms that may have different exponents. For example, $4x+x^2-3+x$ is a polynomial.

To simplify polynomials efficiently, add an extra step: organize the terms in descending order (highest exponent to lowest exponent). This makes like terms easier to spot.

For example:

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

Now combine like terms:

• $4x+x=5x$

So the simplified expression is $x^2+5x-3$. Many quadratic expressions are written in descending order because that form is standard and makes simplification easier.

Try an example. Use this process: 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’ll focus on what factoring is doing.

The purpose of factoring is to find something that every term has in common and pull it out of the expression. You don’t throw that common factor away - you write it outside parentheses to show that it could be distributed back in to recover the original expression.

For instance:

$$2x+4$$

Both terms share a factor of $2$. Factor out $2$:

$$2(x+2)$$

If you distribute the $2$ back into the parentheses, you return to the original expression:

$$2x+4$$

You can also factor out variables. When you factor out $x$, you are dividing each term by $x$, so $x/x=1$. Consider:

$$2x^2+x$$

Factor out $x$:

$$x(2x+1)$$

This skill is especially useful for simplifying fractions. If you can factor an expression so that the numerator and denominator share a common factor, you can cancel that factor.

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)?

FOIL

Sometimes the ACT will give you an expression that is already factored. For example, you might see two binomials (expressions with two terms), such as $(x+3)(x-5)$. To turn this back into a polynomial, you can’t just multiply the first terms. You must multiply every term in the first binomial by every term in the second.

The most common and reliable way to do this is the FOIL method.

FOIL stands for:

• F - First (multiply the first terms)
• O - Outside (multiply the outer terms)
• I - Inside (multiply the inner terms)
• L - Last (multiply the last terms)

Let’s work through an example:

Example: Expand and simplify

$$(x+3)(x-5)$$

Step 1: Apply FOIL

F: Our first terms are $x$ and $x$. Multiply them together to get $x^2$. O: Our outer terms are $x$ and $-5$. Multiply them together to get $-5x$. I: Our inner terms are $3$ and $x$. Multiply them together to get $3x$. L: Our last terms are $3$ and $-5$. Multiply them together to get $-15$.

Step 2: Combine like terms $x^2-5x+3x-15$ $x^2-2x-15$

Done!

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.