 
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 

8x^{3} − 2x^{2} 
Strategy 
8x^{3} and 2x^{2} 
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. 
x^{3} has factors
of x, x^{2}, x^{3}
x^{2} has factors of x, x^{2}
x^{3} and x^{2}
have x^{2} in common. 
Find the common factor for
factors involving a variable 
Hence 2x^{2}
is the largest common factor. 

8x^{3} − 2x^{2}
= 2x^{2}(4x) − 2x^{2}(1)
= 2x^{2}(4x − 1) 
Put the
parts together. 
Have a go at these now.
Now you can use your factorspotting skills to factorise expressions. Here is one
more example to remind you how it works:
Example
2 

4x^{5}  12x^{3} + 8x^{6} 
Strategy 
4x^{5} and 12x^{3}
and 8x^{6} 
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. 
x^{5} has factors
of x, x^{2}, x^{3} x^{4} x^{5}
x^{3} has factors of x, x^{2}, x^{3}
x^{6} has factors of x, x^{2} x^{3} x^{4} x^{5}
x^{6}
x^{5},x^{3}
and x^{6} have x^{3} in common. 
Find the common factor for
factors involving a variable 
Hence 4x^{3}
is the largest common factor. 

4x^{5}  12x^{3} + 8x^{6}
= 4x^{3}(x^{2})  4x^{3}(3)
+ 4x^{3}(2x^{3})
= 4x^{3}(x^{2}  3 + 2x^{3}) 
Put the parts together. 
Now have another
go yourself:
Exercise
2
Enter the largest common factor of the expression
2x^{4} + 6x^{2}  18x 

Click on the other factor which will appear in the factorised expression:
2x^{4} + 6x^{2}  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:
Common factors may also involve
functions such as trigonometric or exponential functions.
Factorisation Index 
Trigonometric Functions >>
