Factorization: Common factor extraction - Examples, Exercises and Solutions

Factorization

The factorization we do by extracting the common factor is our way of modifying the way the exercise is written, that is, from an expression with addition to an expression with multiplication.

For example, the expression
2A+4B2A + 4B
is composed of two terms and a plus sign. We can factor it by excluding the largest common term.
In this case it is 2 2 .

We will write it as follows:
​​​​​​​2A+4B=2×(A+2B)​​​​​​​2A + 4B = 2\times (A + 2B)

Since both terms ( A A and B B ) were multiplied by 2 2 we could "extract" it. The remaining expression is written in parentheses and the common factor (the 2 2 ) is kept out.
In this way we went from having two terms in an addition operation to having a multiplication. This procedure is called factorization.

You can also apply the distributive property to do a reverse process as needed.
In certain cases we will prefer to have a multiplication and in others an addition.

Practice Factorization: Common factor extraction

Exercise #1

Find the common factor:

ab+bc ab+bc

Video Solution

Step-by-Step Solution

ab+bc=a×b+b×c ab+bc=a\times b+b\times c

Let's consider that the common factor is b, so we will remove it:

b(ab+bc)= b(ab+bc)=

We divide by b:b(abb+bcb)= b(\frac{ab}{b}+\frac{bc}{b})=

b(a+c) b(a+c)

Answer

b(a+c) b(a+c)

Exercise #2

Find the biggest common factor:

12x+16y 12x+16y

Video Solution

Step-by-Step Solution

We break down the coefficients 12 and 16 into multiplication exercises with a multiplier factor to then simplify:

3×4×x+4×4×y 3\times4\times x+4\times4\times y

We extract 4 which is the common factor:

4(3×x+4×y)=4(3x+4y) 4(3\times x+4\times y)=4(3x+4y)

Answer

4(3x+4y) 4(3x+4y)

Exercise #3

Find the common factor:

7a+14b 7a+14b

Video Solution

Step-by-Step Solution

We divide 14 in a multiplication exercise to help us simplify accordingly:7×a+7×b×2= 7\times a+7\times b\times2=

We extract the common factor 7:

7(a+2×b)=7(a+2b) 7(a+2\times b)=7(a+2b)

Answer

7(a+2b) 7(a+2b)

Exercise #4

Decompose the following expression into factors:

20ab4ac 20ab-4ac

Video Solution

Step-by-Step Solution

We will break down the coefficient of 20 into a multiplication exercise that will help us simplify:5×4×a×b4×a×c 5\times4\times a\times b-4\times a\times c

We extract 4a as a common factor:4a(5×bc)=4a(5bc) 4a(5\times b-c)=4a(5b-c)

Answer

4a(5bc) 4a(5b-c)

Exercise #5

Find the common factor:

25y100xy2 25y-100xy^2

Video Solution

Step-by-Step Solution

First, we will decompose the coefficients of the multiplication exercise that will help us find the common factor:

25×y4×25×x×y×y 25\times y-4\times25\times x\times y\times y

Now find the common factor 25y:

25y(14xy) 25y(1-4xy)

Answer

25y(14xy) 25y(1-4xy)

Exercise #1

Decompose the following expression into factors:

15a2+10a+5 15a^2+10a+5

Video Solution

Step-by-Step Solution

Factor the given expression:

15a2+10a+5 15a^2+10a+5 We will do this by taking out the greatest common factor, both from the numbers and the letters,

We will refer to the numbers and letters separately, remembering that a common factor is a factor (multiplier) common to all terms of the expression,

Let's start with the numbers

Note that the numerical coefficients of the terms in the given expression, that is, the numbers: 5,10,15 are all multiples of the number 5:

15=3510=25 15=3\cdot\underline{5}\\ 10=2\cdot\underline{5}\\ Therefore, the number 5 is the greatest common factor of the numbers,

For the letters:

Note that only the first two terms on the left depend on x, the third term is a free number that does not depend on x, therefore there is no common factor for all three terms together for the letters (that is, we will consider the number 1 as the common factor for the letters)

Therefore, we summarize:

The greatest common factor (for numbers and letters together) is:

515 5\cdot1\\ \downarrow\\ 5 Let's take it, then, as a multiple outside the parenthesis and ask the question: "How many times will we multiply the common factor (including its sign) obtaining each of the terms of the original expression (including its sign)?", so we will know what is the expression inside the parenthesis that multiplied the common factor:

15a2+10a+553a2+5(+2a)+5(+1)5(3a2+2a+1) \textcolor{red}{ 15a^2}\textcolor{blue}{+10a} \textcolor{green}{+5} \\ \underline{5}\cdot\textcolor{red}{3a^2}+\underline{5}\cdot\textcolor{blue}{(+2a)}+\underline{5}\cdot\textcolor{green}{(+1)}\\ \downarrow\\ \underline{5}(\textcolor{red}{3a^2}\textcolor{blue}{+2a}\textcolor{green}{+1}) In the previous expression, the operation is explained through colors and signs:

The common factor has been highlighted with an underscore, and the multiples inside the parenthesis are associated with the terms of the original expression with the help of colors, note that in the detail of the decomposition above we also refer to the sign of the common factor (in black) that we extracted as a multiple outside the parenthesis and the sign of the terms in the original expression (in colors), there is no obligation to show it. This is in stages as described above, you can (and it is worth) jump directly to the broken down form in the last line, but you definitely should refer to the previous signs, since in each member the sign is an inseparable part of it,

We can ensure that this decomposition is correct easily by opening the parentheses with the help of the distributive property and ensuring that the original expression that we decomposed is effectively obtained back - member, this must be done emphasizing the sign of the members in the original expression and the sign (which is always selectable) of the common factor.

(Initially, you should use the previous colors to ensure you get all the terms in the original expression and belong to the multiple inside the parenthesis; later, it is recommended not to use the colors)

Therefore, the correct answer is option b.

Answer

5(3a2+2a+1) 5(3a^2+2a+1)

Exercise #2

Decompose the following expression into factors:

4a+13b+58c 4a+13b+58c

Video Solution

Step-by-Step Solution

Factor the given expression:

4a+13b+58c 4a+13b+58c We will do this by extracting the greatest common factor, both from the numbers and the letters,

We will refer to the numbers and letters separately, remembering that a common factor is a factor (multiple) common to all terms of the expression,

Let's start with the numbers:

We will notice that the numerical coefficients of the terms in the given expression, that is, the numbers 4, 13, 58, do not have a common factor, and this is because the number 13 is a prime number and the other two numbers are not multiples of it,

Therefore, there is no common factor for the numbers (we consider the number 1 (it is the power of zero), as the common factor for the numbers)

For the letters:

There are three terms in the expression:
a,b,c a,\hspace{4pt}b,\hspace{4pt}c It is easy to see that there is no common factor to these three terms,

Therefore, it is not possible to factor the given expression with the help of the common factor.

Therefore, the correct answer is option d.

Answer

It is not possible to factorize the given expression by extracting the common factor.

Exercise #3

Decompose the following expression into factors by removing the common factor:

xyz+yzt+ztw+wtr xyz+yzt+ztw+wtr

Video Solution

Step-by-Step Solution

Factor the given expression:

xyz+yzt+ztw+wtr xyz+yzt+ztw+wtr
We will do this by extracting the highest common factor, both from the numbers and the letters.

We refer to the numbers and letters separately, remembering that a common factor is a factor (multiplier) common to all terms of the expression.

As the given expression does not have numeric coefficients (other than 1), we will look for the highest common factor of the letters:

There are four terms in the expression:
xyz,yzt,ztw,wtr xyz,\hspace{4pt}yzt,\hspace{4pt}ztw,\hspace{4pt}wtr We will notice that in each of the four members there are three different letters, but there is not one or more letters that are included (in the multiplication) in all the terms; that is, there is no common factor for the four terms and therefore it is not possible to factor this expression by extracting a common factor.

Therefore, the correct answer is option d.

Answer

It is not possible to decompose the given expression into factors by extracting the common factor.

Exercise #4

Decompose the following expression into factors:

14xyz+8x2y3z 14xyz+8x^2y^3z

Video Solution

Step-by-Step Solution

First, we break down all the powers into multiplication exercises and at the same time try to reduce the integers as much as possible:

 

7*2*xyz+2*4*x*x*y*y²*z

Now we use the substitution property to arrange the equation in a slightly more convenient way:

2*x*y*z*7+2*x*y*z*x*y²

Now we try to find the common factor among all the parts - 2xyz

2xyz(7+xy²)

Answer

2xyz(7+4xy2) 2xyz(7+4xy^2)

Exercise #5

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 #1

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 #2

Decompose the following expression into factors:

36mn60m 36mn-60m

Video Solution

Answer

12m(3n5) 12m(3n-5)

Exercise #3

Decompose the following expression into its factors:

26a+65bc 26a+65bc

Video Solution

Answer

13(2a+5bc) 13(2a+5bc)

Exercise #4

Decompose the following expression into factors:

13abcd+26ab 13abcd+26ab

Video Solution

Answer

13ab(cd+2) 13ab(cd+2)

Exercise #5

Decompose the following expression into factors:

37a+6b 37a+6b

Video Solution

Answer

No es posible descomponer los factores de la expresión dada

Topics learned in later sections

  1. Algebraic Method
  2. The Extended Distributive Property