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Simplifying Expressions Using Common Factors

Introduction

The concept of factoring can best be understood as being the opposite of multiplying.  In algebra, a pair of binomials can be multiplied as follows:

Intro_to_Factoring_vis_1

 

 

A factoring problem is the exact opposite of this.  The polynomial 14x3 + 21x  is slightly different than the answer above.  The terms 14x3 and 21x have a common factor of 7x.  Dividing each term by the common factor results in 2x2 + 3 being left as the remaining (second) factor. 

 

Intro_to_Factoring_vis_1a


Lesson

The word factoring means to break down a number or expression into smaller pieces, or factors.  These factors can be multiplied together to obtain the original number.

Intro_to_Factoring_vis_2

 

 

When algebraic expressions, the simplest kind of problems are those where each of the terms in the expression can be reduced by a common factor.

 

 

Intro_to_Factoring_vis_3

 

 

Example 1:  Find a common factor in each expression.  Then simplify the expression.

 

Intro_to_Factoring_vis_4a

Intro_to_Factoring_vis_5a

Intro_to_Factoring_vis_6a

 

*Scroll over the problem to see the answer.

 

The same concept can also be used on expressions with 3 or more terms.  Choose a factor that can be divided into each term to in order to simplify the expression.

 

Example 2:  Find a common factor in each expression containing 3 or more terms.  Then simplify the expression.

 

Intro_to_Factoring_vis_7a

Intro_to_Factoring_vis_8a

Intro_to_Factoring_vis_9a

*Scroll over the problem to see the answer.

 

In future lessons, you will factor more complicated expressions such as

  • 8x3y5 + 4x2y in which each term has a common factor of 4x2y
  • x2 + 6x + 5 which can be done using another factoring method

 

 

The method outlined in this lesson works for problems whose terms contain a single variable and have a common factor.  Other lessons cover the above (bulleted) methods of factoring.

Try It

Find a common factor for each of the terms.  Then factor the expression.

 

1)  3d2 + 9

2)  15e3 + 7e2

3)  4f2 + 20f

4)  3g5 + 3g4 + 3g3

5)  21h3 + 49h

6)  i4 + 3i3 + 11i2

7)  75j4 + 66j3 + 45j2

 

 

 

Scroll down for answers:

 

 

 

 

 

 

 

 

 

 

Answers:

 

1)  3d2 + 9  =  3(d2 + 3)

 

2)  15e3 + 7e2  =  e2(15e + 7)

3)  4f2 + 20f  =  4f(f + 5)

4)  3g5 + 3g4 + 3g3  =  3g3(g2 + g + 1)

5)  21h3 + 49h  =  7h(3h2 + 7)

6)  i4 + 3i3 + 11i2  =  i2(i2 + 3i + 11)

 

7)  75j4 + 66j3 + 45j2  =  3j2(25j2 + 22j + 15)


 

 

Related Links:

Didn't find what you were looking for in this lesson?  More information related to factoring can be found at the following places:


Related Algebra Lessons

 

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