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Factorization

Polynomials

You are probably familiar with the process of "multiplying out", that is to say, using the distributive law to express a product as a sum of terms. For example:  3(x + 2) = 3(x) + 3(2) = 3x + 6.

Sometimes, especially when solving equations, it is useful to reverse the process of "multiplying out", that is to say, to express the sum of a number of terms as a product. This process is called factorisation. There are a number of ways of doing this, but by far the most important in practice is what is called "taking out a common factor".

Essentially, this is just applying the distributive law in reverse. For example, multiplying out gives 3(x + 2) = 3(x) + 3(2) = 3x + 6. Looking at it the other way round, suppose we start with the expression 3x + 6. Notice that since 6 = 3(2), we can write the expression as 3(x) + 3(2), where 3 is a factor of both terms (we call it a common factor of the terms). Now we can apply the distributive law in reverse:

3x + 6 = 3(x) +3(2) = 3(x + 2)

and so we have factorised the expression by taking the common factor 3 and putting it outside the brackets. A factor does not have to be an explicit number, it may be a variable (letter x in this case).

Example 1

 

8x3 − 2x2

Strategy

8x3 and 2x2

Consider each product separately

8 has factors of 1,2,4,8
2 has factors of 1,2

The HCF of 8 and 2 is 2

Find the highest common factor of the numeric parts of the product expressions.

x3 has factors of x, x2, x3
x2 has factors of x, x2

x3 and x2 have x2 in common.

Find the common factor for factors involving a variable

Hence 2x2 is the largest common factor.

 

8x3 − 2x2
= 2x2(4x) − 2x2(1)
= 2x2(4x − 1)

Put the parts together.

Have a go at these now.

Exercise 1

Click on the largest common factor of the expression

12x4 + 3x3

 

 

What is the largest common factor for each of these?

  1. 3x3 + 6x2 - 30x
  2. 2p4 + 4p2
  3. 24z3 + 6z2 + 12z
  4. 4m2 + 10x2 - 2mx

 

Now you can use your factor-spotting skills to factorise expressions. Here is one more example to remind you how it works:

Example 2

 
4x5 - 12x3 + 8x6

Strategy

4x5 and 12x3 and 8x6

Consider each product separately

4 has factors 1,2,4
12 has factors of 1,2,3,4,6,12
8 has factors of 1,2,4,8

The HCF of 4,12 and 8 is 4.

Find the highest common factor of the numeric parts of the product expressions.

x5 has factors of x, x2, x3 x4 x5
x3 has factors of x, x2, x3
x6 has factors of x, x2 x3 x4 x5 x6

x5,x3 and x6 have x3 in common.

Find the common factor for factors involving a variable

Hence 4x3 is the largest common factor.

 

4x5 - 12x3 + 8x6
= 4x3(x2) - 4x3(3) + 4x3(2x3)
= 4x3(x2 - 3 + 2x3)

Put the parts together.

Now have another go yourself:

Exercise 2

Enter the largest common factor of the expression

2x4 + 6x2 - 18x

Click on the other factor which will appear in the factorised expression:

2x4 + 6x2 - 18x

Sometimes a common factor may itself be a sum of terms.

Example 3

 

12(x - 3)5 - 3x + 9

Strategy

Notice that -3x + 9 = -3(x - 3)

Consider 12(x - 3)5 and -3(x - 3)

Consider each product separately, but watch out for sums which may be in common.

12 and -3 have a HCF of 3.

Find the highest common factor of the numeric parts of the product expressions.

(x - 3)5 and (x - 3) have (x - 3) in common.

Find the common factor for factors involving a variable

Hence 3(x - 3) is the largest common factor.

 

12(x - 3)5 - 3x + 9
= 3(x - 3)(x - 3)4 - 3(x - 3)(1)
=3(x - 3)((x - 3)4 - 1)

Put the parts together.

Here are a few like this for you to practice on:

Exercise 3

Factorise these then check your answers.

  1. 4(x + 1) + 6(x + 1)2
  2. 2(x -5)4 + 6x2 - 30x
  3. 2(x + 2)4 + 4(x + 2)2
  4. (x + 2)3 + 6x2 + 12x
  5. (7 - 2x)2 + 10x2 - 35x
  6. 10(2r + s)3 - 8(2r + s)2

 

Common factors may also involve functions such as trigonometric or exponential functions.

 

Factorisation Index |  Trigonometric Functions >>

 

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