Factoring and multiplying out are important techniques for simplifying expressions. You need these procedures, for example, if you want to solve equations or have to calculate zeros. Factoring out and multiplying out will accompany you all the way to your final math exam.
The distributive law – factoring out and multiplying out
Before we get to the actual multiplication and factoring, we need a quick look at the basics.
Do you remember the distributive law? When integers are multiplied together, this rule applies.
The distributive law is THE basis for factoring out and multiplying out. Therefore, if you still feel unsure about it, you should first look at the article «Distributive law».
Factoring out and multiplying out – explanation
Factoring and multiplying out are two equivalent transformations that are performed in terms.
Reminder: an equivalence transformation is a transformation of an equation or term that does not change the solution set or the result.
That Multiply out is a transformation that eliminates one or more parentheses. For example, if you read the distributive law from left to right, you multiply out there.
That factoring out is the inverse of multiplying out. A parenthesis is set here by separating a factor that occurs in several summands. Do you read e.g. B. the distributive property from right to left, is excluded there.
Figure 1: Factor out and multiply out
Factoring out and multiplying out are precisely defined two applications of the distributive law:
With Multiply out denotes an equivalent transformation in which, by applying the distributive law, a product in a total is converted.
With factoring out or factoring denotes an equivalent transformation in which, by applying the distributive law, a total in a product is converted.
Reminder: A product is the result of a multiplication or an expression that represents a multiplication. A sum is the result of an addition or an expression representing a sum. In this definition, sum and product are used as labels of terms.
Multiply – that’s how it works!
Now let’s take a close look at how factoring and multiplying out works. Since multiplying out is a little easier, let’s start with that.
There are several cases where you can multiply out:
- A number or a variable (i.e. a factor) is multiplied by a bracket containing a sum or difference (i.e. a line calculation).
- Two or more brackets, each containing sums or differences, are multiplied with each other.
Multiply out – Case 1
Of the first case is basically the distributive law. If you don’t quite understand the distributive property, then you should definitely read the article again.
Here we look at two short calculation examples.
Multiply out – Case 2
The more interesting and also more complicated case is this second casewhen two or even more brackets are multiplied together.
if two If brackets are multiplied with each other, the procedure is as follows:
Each term in the front bracket is multiplied by each term in the back bracket.
So that you don’t forget any of these connections, it is advisable to draw auxiliary arcs or arrows. Here you can also work with colors in your math exercise book. It may also help you to underline the pairs on the left side of the = in order to keep track. It makes sense at the latest when there are no longer just two brackets, each with two terms!
Figure 2: Help sheets for multiplying out
Figure 3: Help sheets and markings for multiplying out
If you want to understand this rule in detail, you can take a look at the following in-depth information.
Proof of the rule for multiplying two brackets
The second case is basically just the application of the distributive law, but it is used twice in a row.
A term of the form is given. We take the second parenthesis as a part of a term, as if there were simply a number there. We can also define ourselves that way. So be
The term thus changes to and here we apply the distributive law as usual:
Now we put the original parentheses back in for a:
becomes
Now the distributive law is present twice again, once with the triangle and once with the parallelogram. Here you can multiply out again as usual.
That’s exactly what we found out in Figure 2 with the aid of the help sheets.
Now look at the following examples for a better understanding.
Task 1.1
Multiply from:
solution
We proceed according to the rule «each term term in the front bracket is multiplied by each term term in the back bracket».
In a final calculation step, the xy can be combined.
But what if not just two brackets are multiplied with each other, but three or even more?
In principle, you use the procedure for two brackets a few times in a row. You first multiply the first two brackets together and the result, which should then be in brackets again, is multiplied by the next bracket. Proceed in this way until you have worked through all the brackets.
Task 1.2
Multiply from:
solution
First, the two front brackets are multiplied out, and the back one is left for the time being. The best way to do this is to put square brackets around the two brackets that you first multiply with each other.
In the last step, the front bracket was simplified. You should always do this if possible, as it saves you a lot of calculation work!
After the two front brackets have been multiplied with each other, the newly created first bracket can now be multiplied out with the second bracket.
As with most math topics, when multiplying out, practice makes perfect. Therefore, at the end of the article you will find some exercises for factoring out and multiplying out.
Exclude – that’s how it works!
Factoring out or factoring can be a bit more difficult than multiplying out. For this you have to develop the ability to find common factors in the terms.
There are also several cases when factoring out:
- You can one Factor out number or variable.
- You can factor out an entire bracket (or even multiple brackets).
- You can several Factor out numbers or variables.
Whether the first, second or third case applies, you cannot necessarily tell immediately every time. That is why we will show you several examples in the following, in which the different cases apply.
exercise 2
Simplify the given term by factoring out:
solution
In this example, the first case applies: a variable can be excluded. The factor that occurs in all terms is the x. Therefore the x can be factored out.
Not only sums, but also differences can be factored.
task 3
Simplify the following expression by factoring out:
solution
The first case also occurs here, this time a number can be factored out. All terms of the term are divisible by 5:
Thus the factor 5 can be excluded:
task 4
Write the following expression as a product:
solution
The second case occurs here, an entire bracket can be factored out. The parenthesis in this case is .
task 5
Exclude as much as possible:
solution
This example belongs to the more difficult third case. You should first think about what can be excluded.
- An a with an exponent that is at least 2 occurs in all terms. Therefore, one can be excluded.
- All terms are also divisible by 2. The factor 2 can therefore also be factored out.
You can now exclude altogether. It is up to you whether you do this in one step or use two steps by first factoring out one factor and then the other factor. If you don’t feel quite sure yet, it’s better to proceed step by step. Both ways are shown here:
First, step by step, it is factored out:
Here are excluded directly:
To check whether you factored correctly, it helps to carry out a test at the end. In the test, you have to multiply your result again.
To practice factoring out, be sure to look at the practice problems at the end of the chapter.
Multiply out and factor out with fractions
Not only can you factor out whole numbers and variables, but also fractions. This then works in the same way as explained above.
Sometimes it is very useful to factor out fractions to convert an all-fraction expression to an integer expression. But this only works if all fractions have the same denominator, as in the following example.
task 6
Write as product:
solution
All elements of the expression here have a fraction with denominator 7 in common. Therefore it can be excluded:
Even if not all elements have the same denominator, fractions can sometimes be skillfully factored out.
task 7
Cleverly simplify the following expression:
solution
The first thing you might notice is that there is an a in every element. This can be excluded in the first step.
Now the breaks in the back parentheses are still a bit annoying. You can now cleverly expand so that the same denominators appear in the brackets. Here it makes sense to expand to quarters.
Now you can factor out as in the last example.
Factoring out and multiplying out – exercises
Here are some exercises to help you remember what you’ve learned.
task 8
Summarize the following terms as much as possible.
Tip: The specification means that the term should be as clear as possible at the end. Sometimes you have to first multiply out and then factor something out again.
solutions
Solution to 1: You should first multiply out the first term and then combine terms with the same variables.
It is no longer possible to exclude here, since the terms have no common factors.
Solution to 2: Here it makes sense to first multiply out and also to remove the minus brackets.
To remove a minus bracket, all signs in the bracket are reversed.
Example:
Both terms are now divisible by 2. Therefore we can still factor out the factor 2. After that, nothing more is possible.
Solution to 3: Here, too, you cannot avoid multiplying out.
As in the previous exercise, a 2 can now be factored out.
Solution to 4: Caution! You can save yourself the multiplication here if you look very closely. The parentheses are the same in every term and can therefore be factored out as a whole.
task 9
Write as a product.
solutions
Solution to 1: Both terms are divisible by 2. That’s about the only thing that can be excluded.
Solution to 2: In the two terms there is a…