Simplify

Simplification is the process of "simplifying" a mathematical expression, which most often results in the expression being shorter and easier to work with. Simplifying an expression can involve a range of operations such as basic arithmetic, combining like terms, factoring, using exponent or logarithm rules, trigonometric identities, and more.

One of the earliest points we may encounter simplification is learning about fractions. Though simplifying fractions is used within the study of algebra, we won't go into detail about how to simplify fractions here. Refer to the simplifying fractions page for more detail. Instead, we will look at simplification in an algebraic context.

Simplifying polynomial expressions

Polynomial expressions are expressions that only involve the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. If the expression involves division of a variable, it is a rational expression (discussed later), not a polynomial expression.

There are many ways to simplify an expression, but generally, we need to follow order of operations while combining like terms, factoring, or maybe expanding. We can also reduce fractions or terms that share the same factors, whether they are variables or constants. There are numerous tricks or patterns for simplifying that we can notice with practice, and the ability to simplify expressions is a necessary skill within the study of algebra, as well as many other areas of mathematics.

Examples

1. Simplify the expression x + 3x + 4.

x + 3x + 4 = 4x + 4

In this case, simplifying the expression just involved combining the like terms, x and 3x. Depending on the goal, we could also factor the expression 4x + 4.

4x + 4 = 4(x + 1)

In some cases, 4(x + 1) may be considered the simplified form, or if not, it could be preferable to work with this equivalent expression, particularly when solving equations.

2. Simplify (x + 2)2 + 4x - x2 + 3(4 + 2).

Start by adding the terms in parenthesis and expanding (x + 2)2.

x2 + 4x + 4 + 4x - x2 + 3(6)

Then we can start combining like terms.

8x + 4 + 18 = 8x + 22

This is in simplest form, unless we want to have it in factored form, and see that 8 and 22 share a factor of 2:

8x + 22 = 2(4x + 11)

Simplifying rational expressions

The above examples involved polynomial expressions, which for the most part just require basic arithmetic and order of operations. In comparison, rational expressions, which are essentially fractions made up of polynomials (sometimes referred to as algebraic fractions), typically require factoring as part of the simplification process. Like regular fractions, the goal when simplifying rational expressions is to find the greatest common factor (GCF) of the numerator and denominator, after which we can divide both numerator and denominator by the GCF.

Examples

1. Simplify :

When we have variables with the same base raised to an exponent in a rational expression, we can simplify them by subtracting the smaller exponent from the larger one, writing the base wherever the larger exponent was, and raising it to the remainder after subtraction, as we did above. 3 - 2 = 1, and the term in the denominator was larger, so we were left with . Note that the base, in this case x, must be exactly the same to be able to cancel the exponents in this manner.


2. Simplify :

This example involved factoring the greatest common factor out of both the numerator and denominator, then canceling out the largest factor they share, which is 2. We could also write the final denominator as 2x + 6, as we could have recognized that the numerator and denominator shared a factor of 2, and only factored a 2 out of the denominator rather than a 4.


3. Simplify :

This example involved factoring the polynomial in the numerator, then canceling the like terms.