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*Expressing a mathematical expression as a product of its factors is called Factorisation.*

A number can be expressed as product of its factors. Similarly, an algebraic expression can be expressed as a product of its factors. These factors can be numbers, algebraic variables, algebraic terms or algebraic expressions.

A number written as a product of prime factors is said to be in the prime factor form.

When a number is expressed as a product of factors, 1 is not mentioned as a factor, unless it is specially required.

56 = 2×2×2×7

72 = 2×2×2×3×3

### Irreducible form

An algebraic term is a product of its factors. When a term is expressed as a product of its factors which can not be further reduced into factors then the term is said to be in irreducible form.

3*xy *= 3*⋅x⋅y*

4*x*^{2}*y *= 2⋅2*⋅x⋅x⋅y*

21*xy*^{2}*z* = 3⋅7*⋅x⋅y⋅y⋅z*

## Methods of Factorisation

### Method of common factors

➢ Write each term as a product of irreducible factors

➢ Take the common factor out of parenthesis as observed in distributive property

ab +ac = a(b+c)

➢ Combine the terms to remove their irreducible form.

➢ Factor form of an expression has only one term.

12+36*x*

= 2⋅2⋅3 + 2⋅2⋅3⋅3⋅*x*

= 2⋅2⋅3(1+3⋅*x*)

= 12(1+3*x*) (Required factor form)

### Factorisation by regrouping terms

➢ Rearrange the terms in expression to form groups with common factors.

➢ Take out common factors in the groups which will reduce the number of terms in the expression.

➢ Then a common factors can be taken out to lead the factorization further.

➢ Regrouping is possible in more than one way.

*x*^{2} + *x**y *+ 8*x* + 8*y *

= *x*^{2} + 8*x* + *x**y *+ 8y

= *x*(*x*+8) + *y*(*x*+8)

= (*x*+8)(*x*+*y*)

15 *xy *– 6*x *+ 5*y –* 2

= 15 *xy *+ 5*y *– 6*x *– 2

= 5y(3x+1) – 2(3x+1)

= (3x+1)(5y– 2)

### Factorisation using Identities

*(a + b) ^{2}*

*= a*

^{2}+ 2ab + b^{2}*(a − b) ^{2}*

*=*

*a*

^{2}− 2ab + b^{2}*(a + b)(a − b)** = **a*^{2}* − b*^{2}

*(x + a)(x + b)** = x*^{2}* + (a + b)x + ab*

➢ In all these standard identities, the expression on LHS are in factor form.

➢ If we can change the expression to be factorised in a form that fits the RHS of one of the identities, then the expression corresponding to the LHS of the identity can be directly applied to get the desired factorisation.

*a*^{2} + 8*a *+ 16

=*a*^{2} + 2⋅*a*⋅4 + 4^{2}

= (*a*+4)^{2}

*p*^{2} – 10*p* + 25

= *p*^{2 }– 2⋅*p⋅*5 + 5^{2}

= (*p *− 5)^{2}

49x^{2} – 36

= (7x)^{2} – 6^{2}

= (7x+6)(7x–6)

*p*^{2} + 6*p* + 8

=*p*^{2} + 2*p + *4*p* + 2⋅4

= *p*^{2} + (2 + 4)*p* + 2⋅4

= (*p *+ 2)(*p *+ 4)

In general, for factorising an algebraic expression of the type

*x*^{2} + p*x *+ q, we find two factors **a** and **b** of **q** (i.e., the constant term) such that,

ab = q and, a + b = p

Then, the expression becomes

*x*^{2 }+ (a + b) *x* + ab

= *x*^{2} + a*x* + b*x* + ab

= *x*(*x *+ a) + b (*x *+ a)

= (*x* + a) (*x* + b) which are the required factors.

## Division of Algebraic Expressions

➢ Factorize the expressions in the numerator and the denominator.

➢ Cancel the factors common to both the numerator and the denominator.

### Division of a monomial by another monomial

24*x**y*^{2}*z*^{3} ÷ 6y*z*^{2}

= 24*x**y*^{2}*z*^{3} / 6y*z*^{2}

= 2⋅2⋅2⋅3⋅*x⋅y⋅**y⋅**z⋅z⋅z*/ 2⋅3⋅y⋅*z⋅z*

= 2⋅2⋅*x⋅y⋅z*

= 4*xyz*

### Division of a polynomial by a monomial

(3*a*^{8} − 4*a*^{6 }+ 5*a*^{4}) ÷ *a*^{4}

= (3*a*^{8}* − 4a*^{6} +5*a*^{4})/ *a*^{4}

= * a^{4}(*3

*a*

^{4}− 4

*a*

^{2}+5)/

*a*

^{4}

= 3

*a*

^{2}− 4

*a*

^{2 }+ 5

### Division of a polynomial by another polynomial

(*m*^{2} − 14*m* − 32) ÷ (*m *+ 2)

= (*m*^{2} − 14*m* −32)/(*m + 2)*

= (*m*^{2 }+ 2*m –* 16*m * − 2⋅16)/(*m + 2*)

= *m*(*m *+ 2)− 16(*m* + 2)/(*m + 2*)

= (*m *+ 2)( *m *− 16)/(*m *+ 2)

= *m *− 16

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