Learning to simplify algebraic expressions is an essential requirement for mastering basic algebra, as well as being an extremely valuable tool for all mathematicians. Simplification allows a mathematician to make complex, long, or inappropriate expressions into simpler or more convenient forms while still remaining equivalent. The skill of basic simplification is pretty easy to learn - even for those math-averse. By following a few simple steps, it is possible to simplify many of the most common types of algebraic expressions without having any kind of mathematical knowledge. Read Step 1 to get started!
Understanding important concepts
Step 1. Define “related terms” by variables and powers
In algebra, “affine numbers” have the same configuration of variables, being raised to the same powers. In other words, for two terms to be “affine” they must have the same variables, or none at all, and each of them must be raised to the same power, or none at all. The order of variables within the term does not matter.
- For example, 3x2 and 4x2 they are related terms because each of them contains the variable x raised to the second power. However, x and x2 they are not related terms, as each has x raised to a different power. Similarly, -3yx and 5xz are not related terms because they each have a distinct set of variables.
Step 2. Factor in writing numbers as a product of two factors
Factorization is the concept of representing a given number as a product of two factors multiplied together. Numbers can have more than one set of factors - for example, the number 12 can be formed by 1×12, 2×6 and 3×4, so you can declare that 1, 2, 3, 4, 6 and 12 they are all factors of 12. Another way of thinking is that the factors of a number are those numbers by which it is equally divisible.
- For example, if we want to factor 20, we can write it as 4×5.
- Note that variable terms can also be factored. -20x, for example, can be written as 4(-5x).
- Prime numbers cannot be factored because they are only divisible by themselves and 1.
Step 3. Use the acronym PEMDAS to remember the order of operations
Occasionally simplifying an expression means nothing more than performing operations on that expression until that is no longer possible. In such cases, it is important to remember the order of operations so as not to make any arithmetic errors. The acronym PEMDAS can be of great help when you need to remember the order of operations - the letters correspond to the types of operations that must be performed, in order:
Method 1 of 3: Combining Related Terms
Step 1. Write your equation
The simplest algebraic equations, those involving only a few variable terms with integer coefficients and no fractions, radicals, etc., can often be solved in a few steps. As with most math problems, the first step in simplifying the equation is to write it down!
- As an example problem, for the next steps, we will consider the expression 1+2x-3+4x.
Step 2. Identify related terms
Next, search your equation for related terms. Remember that like terms have both the same variables and the same exponents.
- For example, let's identify related terms in the equation 1+2x-3+4x. Both 2x and 4x have the same variable raised to the same exponent (in this case, the x's are not raised to any power). Additionally, 1 and -3 are related terms, as neither has variables. So, in our equation, 2x and 4x and 1 and -3 are related terms.
Step 3. Combine related terms
Now that you've identified related terms, you can combine them to simplify the equation. Add the terms together (or subtract them for negative terms) to reduce each set of terms with variables and exponents equal to a singular term.
- Let's add related terms in our example:
- 2x+4x = 6x.
- 1+(-3) = -2.
Step 4. Create a simplified expression from your simplified terms
After combining your related terms, build an expression from your set of new and simplified terms. You should get a simpler expression, with a term for each different set of variables and exponents in the original expression. This new expression is the same as the first.
- In our example, the simplified terms are 6x and -2, so the new expression will be 6x-2. This simplified expression is the same as the original (1+2x-3+4x), but smaller and easier to solve. It's also simpler to factor, which, as we'll see next, is another important skill in simplification.
Step 5. Obey the order of operations when combining related terms
In extremely simple expressions like the one in the previous example, identifying terms is simple. However, in more complex expressions, such as those involving terms in parentheses, fractions, and radicals, related terms that can be combined may not be readily apparent. In these cases, follow the order of operations, performing operations on the terms in the expression as needed, until only addition and subtraction remain.
- For example, consider the equation 5(3x-1)+x(2x/2)+8-3x. It would be incorrect to immediately identify 3x and 2x as related terms and combine them despite the parentheses, as we must perform other operations first. Initially, we'll perform arithmetic operations on the expression according to the order of operations, in order to get terms we can use. See below:
Now, since only addition and subtraction operations remain, we can combine the related terms
Method 2 of 3: Factoring
Step 1. Identify the greatest common divisor in the expression
Factoring is a way to simplify expressions by removing common factors from expression terms. To start with, find the greatest common divisor that all the terms in the expression share-in other words, the greatest number by which all the terms in the expression are equally divisible.
- Let's use the 9x equation2+27x-3. Note that all terms in the equation are divisible by 3. Since the terms are not equally divisible by another larger number, we can determine that
Step 3. is the greatest common divisor in the expression.
Step 2. Divide the expression terms by the greatest common divisor
Next, divide each term in the equation by the greatest common divisor found. The resulting terms will have lower coefficients than in the original expression.
- Let's factor our equation by its greatest common divisor, 3. To do this, we'll divide each term by 3.
- 9x2/3 = 3x2
- 27x/3 = 9x
- -3/3 = -1
- So our new expression is 3x2+9x-1.
Step 3. Plot your expression as the product of the greatest common divisor and the remaining terms
The new expression is not the same as the previous one, that is, it cannot be said that it is simplified. To make it equal to the previous one, it is necessary to note the fact that it was divided by the greatest common divisor. Enclose your expression in parentheses and set the greatest common divisor of the original equation as the coefficient for the expression in parentheses.
- In the case of our example expression, 3x2+9x-1, we'll close the expression in parentheses and multiply it by the greatest common divisor of the original equation to get 3 (3x2+9x-1). This equation is the same as the original, 9x2+27x-3.
Step 4. Use factorization to simplify fractions
You may now be wondering why factoring is useful if, after removing the greatest common divisor, the new expression must be multiplied by it again. In fact, factorization allows a mathematician to perform a number of tricks when simplifying an expression. One of the simplest involves taking advantage of the fact that multiplying the numerator and denominator of a fraction by the same number will yield an equivalent fraction. See below:
- Let's say our original example expression, 9x2+27x-3, be the numerator of a larger fraction with 3 in its denominator. This fraction would look like this: (9x2+27x-3)/3. We can use factorization to simplify this fraction:
- We replace the factored form of our original expression with the expression in the numerator: [3(3x2+9x-1)]/3.
- Note that now both numerator and denominator share coefficient 3. By dividing both by 3, we get: (3x3+9x-1)/1.
- Since every fraction that has "1" in its denominator is equal to the terms in the numerator, we can say that the original fraction can be simplified to 3x2+9x-1.
Method 3 of 3: Applying Additional Simplification Skills
Step 1. Simplify fractions by dividing common factors
As noted above, if the numerator and denominator of an expression share factors, those factors can be removed entirely from the fraction. Sometimes this will require factoring the numerator, denominator, or both (as was the case described above), while at other times the shared factors will be readily apparent. Note that it is also possible to divide the numerator terms by the expression in the denominator, individually, to obtain a simplified expression.
- Let's take an example that doesn't necessarily require immediate factorization. In the case of fraction (5x2+10x+20)/10, we can maybe divide each term in the numerator by the number 10 in the denominator in order to simplify it, although the coefficient “5” in 5x2 is not greater than 10 and therefore cannot have 10 as a divisor.
- Doing so brings us to the result [(5x2)/10]+x+2. If we prefer, we can rewrite the first term by (1/2)x2 to get the result (1/2)x2+x+2.
Step 2. Use square factors to simplify radicals
Expressions under the square root symbol are called radical expressions. They can be simplified by identifying square factors (factors that are squares of a given number) and performing the square root operation on them separately in order to remove them from under the square root sign.
- Let's take the following example: √(9). If we think of the number 90 as a product of two of its factors, 9 and 10, we can take the square root of 9 to get the integer 3 and remove it from the radical. In other words:
Step 3. Add exponents by multiplying two exponential terms; subtract them by dividing these terms
Some algebraic expressions require the multiplication or division of exponential terms. Instead of computing each exponential term and multiplying or dividing manually, simply add exponents when multiplying and subtract them when dividing, to save time. This concept can also be used to simplify variable expressions.
- For example, consider the expression 6x3×8x4+(x17/x15). On each occasion where it is necessary to multiply or divide by exponents, we will subtract or add, respectively, in order to quickly find a simplified term. See below:
- The reason this works is as follows:
- Multiplying exponential terms is, in essence, like multiplying long strings of non-exponential terms. For example, since x3 = x×x×x and x5 = x×x×x×x×x, x3×x5 = (x×x×x)×(x×x×x×x×x), or x8
- Similarly, splitting exponential terms is like splitting long strings of non-exponential terms. x5/x3 = (x×x×x×x×x)/(x×x×x). Since each term in the numerator can be canceled by a combining term in the denominator, we are left with two x in the numerator and none in the denominator, getting the answer x2.
- Always remember that you must think of these numbers as having plus or minus signs. Many people have a hard time thinking “What sign should I put here?”
- Ask for help when needed!
- Simplifying algebraic expressions is not easy, but once you get the hang of it, you will make use of this skill throughout your life.
- Always look for related terms and don't be fooled by exponents.
- Don't accidentally add up any number, exponent, or operation that doesn't belong in the expression.