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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
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.
3xy = 3⋅x⋅y
4x2y = 2⋅2⋅x⋅x⋅y
21xy2z = 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.
= 2⋅2⋅3 + 2⋅2⋅3⋅3⋅x
= 12(1+3x) (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.
x2 + xy + 8x + 8y
= x2 + 8x + xy + 8y
= x(x+8) + y(x+8)
15 xy – 6x + 5y – 2
= 15 xy + 5y – 6x – 2
= 5y(3x+1) – 2(3x+1)
= (3x+1)(5y– 2)
Factorisation using Identities
(a + b)2 = a2 + 2ab + b2
(a − b)2 = a2 − 2ab + b2
(a + b)(a − b) = a2 − b2
(x + a)(x + b) = x2 + (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.
a2 + 8a + 16
=a2 + 2⋅a⋅4 + 42
p2 – 10p + 25
= p2 – 2⋅p⋅5 + 52
= (p − 5)2
49x2 – 36
= (7x)2 – 62
p2 + 6p + 8
=p2 + 2p + 4p + 2⋅4
= p2 + (2 + 4)p + 2⋅4
= (p + 2)(p + 4)
In general, for factorising an algebraic expression of the type
x2 + px + 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
x2 + (a + b) x + ab
= x2 + ax + bx + 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
24xy2z3 ÷ 6yz2
= 24xy2z3 / 6yz2
= 2⋅2⋅2⋅3⋅x⋅y⋅y⋅z⋅z⋅z/ 2⋅3⋅y⋅z⋅z
Division of a polynomial by a monomial
(3a8 − 4a6 + 5a4) ÷ a4
= (3a8 − 4a6 +5a4)/ a4
= a4(3a4 − 4a2 +5)/ a4
= 3a2 − 4a2 + 5
Division of a polynomial by another polynomial
(m2 − 14m − 32) ÷ (m + 2)
= (m2 − 14m −32)/(m + 2)
= (m2 + 2m – 16m − 2⋅16)/(m + 2)
= m(m + 2)− 16(m + 2)/(m + 2)
= (m + 2)( m − 16)/(m + 2)
= m − 16