Extracting the common factor in parentheses - Examples, Exercises and Solutions

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 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.

Suggested Topics to Practice in Advance

  1. Factoring using contracted multiplication

Practice Extracting the common factor in parentheses

Exercise #1

2x904x89=0 2x^{90}-4x^{89}=0

Video Solution

Step-by-Step Solution

The equation in the problem is:

2x904x89=0 2x^{90}-4x^{89}=0 Beginning Let'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 2x89 2x^{89} and that is because 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 letters,

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 if so and performing the factorization:

2x904x89=02x89(x2)=0 2x^{90}-4x^{89}=0 \\ \downarrow\\ 2x^{89}(x-2)=0 Let's continue and refer to the fact 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, so, since the only way to get the result 0 from a product is for at least, one of the factors in the product on the left side, must be equal to zero,

Meaning:

2x89=0/:2x89=0/89x=0 2x^{89}=0 \hspace{8pt}\text{/}:2\\ x^{89}=0 \hspace{8pt}\text{/}\sqrt[89]{\hspace{6pt}}\\ \boxed{x=0} In the solution of the equation above, we first equated the two sides of the equation by moving the term with the unknown and then we took out a root of order 89 on both sides of the equation.

(In this case taking out an odd-order root on the right side of the equation yields one possibility)

Or:

x2=0x=2 x-2=0 \\ \boxed{x=2}

In summary if so the solution to the equation:

2x904x89=02x89(x2)=02x89=0x=0x2=0x=2x=0,2 2x^{90}-4x^{89}=0 \\ \downarrow\\ 2x^{89}(x-2)=0 \\ \downarrow\\ 2x^{89}=0 \rightarrow\boxed{ x=0}\\ x-2=0\rightarrow \boxed{x=2}\\ \downarrow\\ \boxed{x=0,2} And therefore the correct answer is answer a.

Answer

x=0,2 x=0,2

Exercise #2

Extract the common factor:

4x3+8x4= 4x^3+8x^4=

Video Solution

Step-by-Step Solution

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

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

x4=x3x 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 4 and for the letters it is:

x3 x^3 and therefore for the extraction

4x3 4x^3 outside the parenthesis

We obtain the expression:

4x3+8x4=4x3(1+2x) 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.

Answer

4x3(1+2x) 4x^3(1+2x)

Exercise #3

Solve the following by removing a common factor:

6x69x4=0 6x^6-9x^4=0

Video Solution

Step-by-Step Solution

First, we take out the smallest power

6x69x4= 6x^6-9x^4=

6x4(x21.5)=0 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 0

6x4=0 6x^4=0

We divide by: 6x3 6x^3

x=0 x=0

x21.5=0 x^2-1.5=0

x2=1.5 x^2=1.5

x=±32 x=\pm\sqrt{\frac{3}{2}}

Answer

x=0,x=±32 x=0,x=\pm\sqrt{\frac{3}{2}}

Exercise #4

x4+2x2=0 x^4+2x^2=0

Video Solution

Answer

x=0 x=0

Exercise #5

x2x=0 x^2-x=0

Video Solution

Answer

x=0,1 x=0,1

Exercise #1

7x3x2=0 7x^3-x^2=0

Video Solution

Answer

x=0,x=17 x=0,x=\frac{1}{7}

Exercise #2

7x1014x9=0 7x^{10}-14x^9=0

Video Solution

Answer

x=2,x=0 x=2,x=0

Exercise #3

x4+x2=0 x^4+x^2=0

Video Solution

Answer

x=0 x=0

Exercise #4

x6+x5=0 x^6+x^5=0

Video Solution

Answer

x=1,x=0 x=-1,x=0

Exercise #5

x64x4=0 x^6-4x^4=0

Video Solution

Answer

x=0,x=±2 x=0,x=\pm2

Exercise #1

x54x4=0 x^5-4x^4=0

Video Solution

Answer

x=4,x=0 x=4,x=0

Exercise #2

3x2+9x=0 3x^2+9x=0

Video Solution

Answer

x=0,x=3 x=0,x=-3

Exercise #3

4x412x3=0 4x^4-12x^3=0

Solve the equation above for x.

Video Solution

Answer

x=0,3 x=0,3

Exercise #4

x7x6=0 x^7-x^6=0

Video Solution

Answer

x=0,1 x=0,1

Exercise #5

x75x6=0 x^7-5x^6=0

Video Solution

Answer

x=0,5 x=0,5

Topics learned in later sections

  1. Factorization
  2. Factorization: Common factor extraction
  3. Factoring Trinomials
  4. Algebraic Fractions
  5. Simplifying Algebraic Fractions
  6. Factoring Algebraic Fractions
  7. Addition and Subtraction of Algebraic Fractions
  8. Multiplication and Division of Algebraic Fractions
  9. Solving Equations by Factoring
  10. Uses of Factorization