Extracting the common factor in parentheses

Common Factor Extraction Method:
Identify the largest free number that we can extract.
Then, let's move on to the variables and ask what is the least number of times the $X$ appears in any element?
Multiply the free number by the variable the same number of times we have found and we will obtain the greatest common factor.

To verify that you have correctly extracted the common factor, open the parentheses and see if you have returned to the original exercise.

Detailed explanation

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Test yourself on factorization of common factor out of parenthesis!

$$ x^2-x=0 $$

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Factoring out the greatest common factor is the first operation we try to carry out when we want to break down an expression into factors.
We can remove from the parentheses a factor that is common to both elements and leave inside a simple and comfortable expression.
The greatest common factor is the largest factor that is completely common to both elements.

Operation steps for extracting the common factor

Notice which is the largest free number we can extract.
Then, let's move on to the unknowns and ask what is the least number of times that $X$ appears in any element?
Multiply the free number by the unknown the number of times we have found and we will obtain the greatest common factor. To verify that you have extracted the common factor correctly, open the parentheses and see if you have arrived at the original exercise.

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Test your knowledge

Question 1

$$ x^4+2x^2=0 $$

Question 2

$$ x^7-x^6=0 $$

Question 3

$$ 4x^4-12x^3=0 $$

Solve the equation above for x.

Let's look at an example of factoring out the common factor from the parentheses.

$8x^2+4x=$
Let's see what is the greatest common factor we can take out, the answer is $4$ .
Now let's move on to the variable $x$ . What is the minimum number of times that $x$ appears in any term? The answer is $1$ .
Now let's multiply the common factor obtained by the variable the number of times we have found and it will give us the greatest common factor.
That is: $4\times x=4x$
Let's take out $4x$ as the common factor and we will obtain:
$4x(2x+1)$
This is our factorization.
When we want to find the solutions we will compare it with $0$ and it will give us:
$4x(2x+1)=0$
$X=0$
$2x+1=0$
$2x=-1$
$x=-1/2$
Therefore:
$x=0,-1/2$

Don't worry, as you practice extracting the common factor you will not need to act according to the operation steps as we taught them, you will do the extraction of the common factor intuitively and quickly.

If you are interested in this article, you might also be interested in the following articles

Factorization
The uses of factorization
Factorization according to short multiplication formulas
Factorization of trinomials
Factorization of algebraic fractions
Addition and subtraction of algebraic fractions
Simplification of algebraic fractions
Multiplication and division of algebraic fractions
Solving equations through factorization

In the Tutorela blog, you will find a variety of articles on mathematics.

Examples and exercises with solutions on factoring by taking out the common factor from the parentheses

examples.example_title

Extract the common factor:

$4x^3+8x^4=$

examples.explanation_title

First, we use the power law to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$ It is necessary to keep in mind that:

$x^4=x^3\cdot x$ Next, we return to the problem and extract the greatest common factor for the numbers separately and for the letters separately,

For the numbers, the greatest common factor is

$4$ and for the letters it is:

$x^3$ and therefore for the extraction

$4x^3$ outside the parenthesis

We obtain the expression:

$4x^3+8x^4=4x^3(1+2x)$ To determine what the expression inside the parentheses is, we use the power law, our knowledge of the multiplication table, and the answer to the question: "How many times do we multiply the common factor that we took out of the parenthesis to obtain each of the terms of the original expression that we factored?

Therefore, the correct answer is: a.

It is always recommended to review again and check that you get each and every one of the terms of the expression that is factored when opening the parentheses (through the distributive property), this can be done in the margin, on a piece of scrap paper, or by marking the factor we removed and each and every one of the terms inside the parenthesis, etc.

examples.solution_title

$4x^3(1+2x)$

examples.example_title

Solve the following by removing a common factor:

$6x^6-9x^4=0$

examples.explanation_title

First, we take out the smallest power

$6x^6-9x^4=$

$6x^4\left(x^2-1.5\right)=0$

If possible, we reduce the numbers by a common factor

Finally, we will compare the two sections with: $0$

$6x^4=0$

We divide by: $6x^3$

$x=0$

$x^2-1.5=0$

$x^2=1.5$

$x=\pm\sqrt{\frac{3}{2}}$

examples.solution_title

$x=0,x=\pm\sqrt{\frac{3}{2}}$

Do you know what the answer is?

Question 1

$$ 3x^2+9x=0 $$

Question 2

$$ x^5-4x^4=0 $$

Question 3

$$ x^6-4x^4=0 $$

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