How to find the common factor. Resolved exercises.

Next I’ll show you how to get a common factor, which basically consists of undoing the multiplication again, that is, starting from the result of the multiplication and having a monomial polynomial again. Don’t worry, I’ll explain it very slowly now.

What is common factor extraction

To get a common factor consists of separating the element that is common in each term from an expression, that is, repeating it in all the terms and placing it multiplying it to the terms in which it was initially.

It is the inverse operation of multiplying a monomial (polynomial of a term) by a polynomial of two or more terms.

Let’s look at this concept step by step with an example. We’re going to do this multiplication:

 

We will call step 1 this step

To make this multiplication, we multiply the x for each of the terms that we have in the parentheses and we have:

 

We will call step 2 this step.

So far, all normal. I haven’t told you anything new.

Now imagine we’re starting from step 2:

And we want to get to step 1:

Moving from step 2 to step 1 is what is called getting or taking out common factor.

In the first term of step 2, we have x², i. e. we have the x multiplied twice and in the second term we have the x repeated only once.

By finding a common factor, we remove one x from each term and put it in front of the two terms by multiplying (now with one x less each):

 

Sometimes it is necessary to perform the operation inverse to this multiplication, which is what we call taking the common factor. It is used to solve incomplete second-degree equations, factor polynomials or to simplify algebraic fractions.

Once we understand the concept, we will see step by step how to take common ground.

How to get the common factor of polynomials step by step

To get the common factor of polynomials, we must take into account what the common factor means:

  • Factor: is multiplying the rest of the term
  • Common: repeated in all terms

And therefore, in order to be able to get a common factor, the factor to be drawn must compulsorily fulfil these two premises.

Let’s see with a very simple example how to get common factor:

 

The first step is to identify what the common factor is, the factor that repeats itself in all terms.

In this case it’s the x, since it fulfills both conditions: it multiplies the rest of the term and is repeated in both terms:

 

Once we’ve identified the common factor, we put it in front of us and open parentheses:

What we have to do now is divide the polynomial between the common factor:

We have a polynomial in the numerator and a monomial in the denominator. Therefore, we must divide each of the terms of the polynomial between that monomial (taking into account the sign in front of them). In other words, we have to divide the monomial.

First polynomial term between the common factor:

 

Second polynomial term between the common factor:

Each of these results are in parentheses (with its corresponding sign), multiplied by the common factor:

Let’s look at it with another example:

In this case we see that the first term has 2 factors x (x²=x. x) and the second term has an x.

Therefore an x is repeated in both terms, so x is the common factor. We put it in front of us multiplying and we open parentheses:

 

Now we have to divide the polynomial between the common factor:

First polynomial term between the common factor:

Second polynomial term between the common factor:

And the result is enclosed in parentheses, multiplied by the x:

A mistake that is usually made quite often is to put a 0 in the second term. Remember that we divide each term between x, so in the second term there is a 1 (which is x/x) and not a 0 (which would be x-x).

To check if you have taken the common factor correctly, just multiply it again and see if the result is equal to the initial polynomial.

 

How to get common factor in more complex polynomials

The common factor need not always be a single factor.

Factors that are multiplying to the rest of the term and are common to several terms can be taken out as a common factor.

For example:

 

In this case, we see that we can take an x from a common factor, since it is repeated in the three terms, but also a 2 is repeated, since in the second term 6=2.3 and in the third term 4=2.2.

So, the common factor in this case is 2x, which we put in front of it by multiplying and opening parentheses:

 

Now we divide each term of the polynomial between the common factor.

First polynomial term between the common factor:

Second polynomial term between the common factor:

Third polynomial term between the common factor:

And we place each result in parentheses multiplied by 2x:

Let’s look at another example:

In this case I can use a 4 as a common factor, since in the first term I have it directly and in the second term I have an 8 which is 4.2.

I can also get an x and a y.

The common factor will then be 4xy:

 

I divide each term of the polynomial by 4xy. First term of the polynomial:

And second term of the polynomial:

And in the end, I’m left with:

Resolved exercises about taking common factor

1 – Obtain common factor in the following polynomials:

Solution

 

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