What is Factorisation?
We know that to factorise an algebraic expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.
Expressions like 3xy,
On the other hand, consider the expressions- 2x + 4, 3x + 3y,
Method of common factors
Factorise 2x + 4 (Write each term as a product of irreducible factors)
We can write 2x =
and 4 =
x
Therefore we get,
2x + 4 = (2 x) + (2×2)
We notice that the factor 2 is common to both the terms. Using the
2(x + 2) = (2×x) + (2×2) (for the above case)
Therefore, we can write:
2x + 4 = 2×(x + 2) = 2(x + 2)
Thus, the expression 2x + 4 is equal to 2 (x + 2). In this form we can easily see the factors for the expression: 2 and (x + 2). These factors as we know, are
- Factorise 5xy + 10x and find the common factor between the two terms
- The common factor :
- Factorised form =
- Taking out the common factor
- Which gives us the above answer
Therefore, 5xy + 10x = 5 x (y + 2) which is the factorised form
Let's Solve
- Factorise the following expressions:
(A)
- The common factor(s) product :
- Thus, factorised form =
- Taking out the common factor
- Which gives us the above answer
(B)
- The common factor(s) product :
- Thus, factorised form =
- Taking out the common factor
- Which gives us the above answer
(C)
- The common factor(s) product :
- Thus, factorised form =
- Taking out the common factor
- Which gives us the above answer
(D)
- The common factor(s) product :
- Thus, factorised form =
- Taking out the common factor
- Which gives us the above answer
Factorisation by regrouping terms
Take the expression:
2xy + 2y + 3x + 3 as an example
What do we observe? Is there something different about this expression from the ones that we have considered upto this point?
We observe that the terms all together don't share a common factor. The first two terms have the common factors of 2 and y while the last two terms have a common factor of 3. So, how to solve this expression now?
Let's break the expression into smaller pieces and let's just consider the terms '2xy' and '2y'
Writing the term (2xy + 2y) in the factor form, we get:
2xy + 2y =
Similarly we can take the terms (3x + 3) which will give us:
3x + 3 =
Hence combining the results we can write:
2xy + 2y + 3x + 3 =
We notice that the term (x + 1) is common to both the terms on the right hand side of the equation. Upon bringing out the common term we will get:
2xy + 2y + 3x + 3 = 2y (x + 1) + 3 (x + 1)
2xy + 2y + 3x + 3 =
The expression 2xy + 2y + 3x + 3 is now in the form of the product of its factors with the factors being (x + 1) and (2y + 3). Note that these factors are also irreducible.
What is regrouping?
As we just saw in the above expression:
2xy + 3 + 2y + 3x
it is not immediately evident what the possible factors may be. In order to further simplify this process we will rearrange the expression as
2xy + 2y + 3x + 3
thus, allowing us to form the groups (2xy + 2y) and (3x + 3) leading to factorisation. This is known as regrouping.
Regrouping may be possible in more than one ways. Suppose, we regroup and rewrite the expression as:
2xy + 3x + 2y + 3
This will also lead to us getting the factors:
2xy + 3x + 2y + 3 = x × (
This gives us
=
Notice that the factors are the same (as they should be) even though they appear in different order.
Let's Solve
- Find the common factors of the given terms:
- Factorise the following expressions:
(a)
- Take out the common factor(s) product :
- Thus, factorised form =
- Taking out the common factor out of the bracket
- Which gives us the above answer
(b)
- Take out the common factor(s) product :
- Thus, factorised form =
- Taking out the common factor out of the bracket
- Which gives us the above answer
(c)
- Take out the common factor(s) product :
- Thus, factorised form =
- Taking out the common factor out of the bracket
- Which gives us the above answer
(d)
- Take out the common factor(s) product :
- Thus, factorised form =
- Taking out the common factor out of the bracket
- Which gives us the above answer
(e)
- Take out the common factor(s) product :
- Thus, factorised form =
- Taking out the common factor out of the bracket
- Which gives us the above answer
(f)
- We can see that the factorization needs re-grouping.
- Regrouping the expression
- Take out the common factors from the corresponding terms
- Thus, factorised form =
- Which gives us the above factorised answer
(g)
- We can see that the factorization needs re-grouping.
- Regrouping the expression
- Take out the common factors from the corresponding terms
- Thus, factorised form =
- Which gives us the above factorised answer
(h)
- We can see that the factorization needs re-grouping.
- Regrouping the expression
- Take out the common factors from the corresponding terms
- Thus, factorised form =
- Which gives us the above factorised answer
(i)
- The factorization doesn't need re-grouping.
- Take out the common factors from the corresponding terms
- Thus, factorised form =
- Which gives us the above factorised answer
Factorisation using identities
When dealing with expressions, you might have observed that the form of the expressions we encounter are often repetitive. There are a good number of expressions which have a standard form of factorisation which have been proven over time. These
(a + b)2 = a2 + b2 + 2ab (I)
(a - b)2 = a2 + b2 - 2ab (II)
(a + b)(a – b) = a2 - b2 (III)
Therefore, if the expression at hand, has a form that fits the RHS of any one of these identities, then the expression corresponding to the LHS of the identity is the desired factorisation.
Let's Solve
- Factorise the following using identities:
(a)
- The above is in the form of the identity:
where a and b are two integers. - We see that a = x while b =
- Thus, the factorised form =
- Which gives us the above factorised answer
(b)
- The above is in the form of the identity:
where a and b are two integers. - We see that a =
while b = 3 - Thus, the factorised form =
- Which gives us the above factorised answer
(c)
- The above is in the form of the identity:
where a and b are two integers. - We see that a =
while b = 6 - Thus, the factorised form =
- Which gives us the above factorised answer
(d)
- In the above expression: the first three terms are in the form of the identity:
where a and b are two integers. - Factorising the first three terms, we get:
- The resulting expression is in the form:
where a and b are random integers (not the ones mentioned in the current problem). - Thus, the resulting factorised form becomes
- Which gives us the above factorised answer
(e)
- Notice that:
m 4 = and 256 =m 2 2 squared. - Thus, the above expression is in the form of the identity:
where a and b are two integers. - Doing this, we get:
- We see that the expression
can be further factorised using the same identity.m 2 − 16 - Thus, the resulting factorised form becomes
- Which gives us the above factorised answer
Factors of the form (x + a) (x + b)
Upto now the expressions we have looked at consist of mostly perfect squares i.e.
So, how can these expressions be factorized? One type that may be possible is the type of
The identity is:
(x + a) (x + b) = x 2
- (a + b) x + ab
(x + a)(x + b) = x2 + (a+b)x + ab (IV)
Let's try to use this identity for factorization as we have done earlier.
Factorise
Upon comparing the above with the Identity (IV), we get:
ab = 6 and a + b = 5
From here, we can obtain a and b to give us the factorisation (x + a)(x + b).
Using Trial and Error Method:
Since ab = 6, then a and b must be factors of 6.
If we try a = 6, b = 1 we get:
a + b = 7 (not 5)
So, these values are not right.
If we try a = 2, b = 3:
a + b = 5, giving us the required value.
Thus,
The factorised form is
In general, for factorising an algebraic expression of the type
we try to find two factors a and b of q (the constant term) such that
ab = q and a + b = p
Then, the expression becomes -
=
= x(x + a) + b(x + a)
= (x + a) (x + b)
which gives the required factors of a and b.
Let's Solve
- Factorise the following expressions using the identities:
(1)
- The above identity is in the form of the identity:
where a and b are two integers. - Thus, the resulting factorised form becomes
- So, we get the following
- Which gives us the above factorised answer
(2)
- The above identity is in the form of the identity:
where a and b are two integers. - Thus, the resulting factorised form becomes
- So, we get the following
- Which gives us the above factorised answer
(3)
- Expand the first term using the identity
- Upon simplifying, the resulting factorised form becomes
- So, we get the following
- Which gives us the above factorised answer
(4)
- Upon simplifying using the identity, the resulting factorised form becomes
- So, we get the following
- Which gives us the above factorised answer
(5)
- Take out the common factor of
which gives us the form: a 2 − b 2 - Upon simplifying using the identity, the resulting factorised form becomes
- So, we get the following
- Which gives us the above factorised answer
(6)
- Take out the common factor of
which gives us the form: a 2 − b 2 - Upon simplifying using the identity, the resulting factorised form becomes
- So, we get the following
- Which gives us the above factorised answer
(7)
- Upon expanding and simplifying using the identity, the resulting factorised form becomes
- So, we get the following
- Which gives us the above factorised answer
(8)
- Upon simplifying using the identities, the resulting factorised form becomes
- So, we get the following
- Which gives us the above factorised answer
(9)
- We observe that the
three terms are together form an identity. - Upon simplifying using the identities, the resulting factorised form becomes
- So, we get the following
- Further on
- Which gives us the above factorised answer
(10)
- Upon simplifying using the identities, the resulting factorised form becomes
- So, we get the following as the factorised answer, upon taking the common factor.
(11)
- Upon simplifying using the identities, the resulting factorised form becomes
- Further on
- So, we get the following as the factorised answer
(12)
- Upon simplifying using the identities, the resulting factorised form becomes
- So, we get the following as the factorised answer
(13)
- Upon simplifying using the identities, the resulting factorised form becomes
- Upon simplifying
- We get the factorised answer
(14)
- Upon simplifying using the identities, the resulting factorised form becomes
- Upon simplifying
- We get the factorised answer
(15)
- Upon simplifying using the identities, the resulting factorised form becomes
- Upon simplifying
- Further expanding using identities
- More over we get
- We get the factorised answer
(16)
- Upon simplifying using the identities, the resulting factorised form becomes
- Upon simplifying
- Further expanding using identities
- More over we get
- We get the factorised answer
(17)
- Upon simplifying using the identities, the resulting factorised form becomes
- Upon simplifying
- Further expanding using identities
- More over we get
- We get the factorised answer