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 theparentheses and see if you have returned to the original exercise.
Examples with solutions for Extracting the common factor in parentheses
Exercise #1
2x90−4x89=0
Video Solution
Step-by-Step Solution
The equation in the problem is:
2x90−4x89=0Let's pay attention to the left side:
The expression can be broken down into factors by taking out a common factor, The greatest common factor for the numbers and letters in this case is 2x89since the power of 89 is the lowest power in the equation and therefore included both in the term where the power is 90 and in the term where the power is 89.
Any power higher than that is not included in the term where the power of 89 is the lowest, and therefore it is the term with the highest power that can be taken out of all the terms in the expression as a common factor for the variables.
For the numbers, note that the number 4 is a multiple of the number 2, so the number 2 is the greatest common factor for the numbers for the two terms in the expression.
Continuing and performing the factorization:
2x90−4x89=0↓2x89(x−2)=0Let's continue and remember that on the left side of the equation that was obtained in the last step there is an algebraic expression and on the right side the number is 0.
Since the only way to get the result 0 from a product is for at leastone of the factors in the product on the left side to be equal to zero,
Meaning:
2x89=0/:2x89=0/89x=0
Or:
x−2=0x=2
In summary:
2x90−4x89=0↓2x89(x−2)=0↓2x89=0→x=0x−2=0→x=2↓x=0,2And therefore the correct answer is answer a.
Answer
x=0,2
Exercise #2
Solve the following by removing a common factor:
6x6−9x4=0
Video Solution
Step-by-Step Solution
First, we take out the smallest power
6x6−9x4=
6x4(x2−1.5)=0
If possible, we reduce the numbers by a common factor
Finally, we will compare the two sections with: 0
6x4=0
We divide by: 6x3
x=0
x2−1.5=0
x2=1.5
x=±23
Answer
x=0,x=±23
Exercise #3
Extract the common factor:
4x3+8x4=
Video Solution
Step-by-Step Solution
First, we use the power law to multiply terms with identical bases:
am⋅an=am+nIt is necessary to keep in mind that:
x4=x3⋅xNext, 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
4and for the letters it is:
x3and therefore for the extraction
4x3outside the parenthesis
We obtain the expression:
4x3+8x4=4x3(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.