An algebraic expression can be written as the product of its factors. Consider an expression, 7yz +3y. We see that 7yz can be written as 7 × y × z. The factors 7,y, and z can not be further expressed as product of factors. So, we can conclude that 7 × y × z is an irreducible form of 7yz.
Expressing the algebraic expression as a product of its factors is called factorization. Factors can be represented in form of numbers, variables or algebraic expressions.
Techniques of Factorization
Let us understand four ways of factorization
Technique of Common Factors
In this method, each term is written as a product of irreducible factors. Then the common factors are identified and expressed using the distributive law.
Example: Factorize 18y2z +21 yz2
18y2z = 2 × 3 × 3 × y × y × z
21yz2 = 3 ×7 × y × z × z
Common factors : 3, y, z
18y2z +21 zy2 = (2 × 3 × 3 × y × y × z) + (3 ×7 × y × z × z)
= 3 × y × z (6 × y + 7 × z)
= 3 y z(6y + 7z)
Technique of regrouping terms
Many times there is no common factor to all terms of the expression. In such a situation, the terms are regrouped until the factors reach the state of being irreducible.
Example: Factorize 16mn – 12m +12n – 9
It is clear that in this expression, there are no common factors among all the terms. We need to regroup the terms.
16mn-12m = 4m(4n-3)
12n-9 = 3 (4n-3)
So, 16mn – 12m +12n -9 = 4m(4n-3)+3 (4n-3)
We can take 4n-3 common between these terms
= (4n-3) (4m+3)
Technique of using Identities
We can make use of identities for factorization.
(a + b)2 = a2 + 2ab +b2 ——–(I)
(a-b)2 = a2 + b2 – 2ab ———-(II)
(a + b) (a- b)= a2 -b2 ———–(III)
Example 1: Factorize 25 x2 – 49
25 x2 – 49 = (5x)2 – (7)2
Using (III), we can rewrite it as
Example 2. Factorize m2 + 36 + 12m
m2 + 36 + 12m
= m2 + 2(6)(m)+62
Using (I) it can be written as
Technique to use the form (x +a) (x +b) as x2 + (a + b)x + ab
We know that x2 + (a + b)x +ab = (x +a)(x+ b)
Example: Factorize m2 +3m+2
Here, we need to find the factors such that their sum is 3 while their product is 2.
These are 2,1
So, m2+ 3m + 2 = m2 + 2m + m +2
= m(m+2) + 1(m+2)
Division of Algebraic Expressions
Division is the inverse of multiplication. Before discussing division, let us recapitulate some terms associated with polynomials. Depending on the number of terms in an expression, we decide if it is a monomial, binomial or a polynomial.
Monomial contains one term while a binomial contains two terms. Generally speaking, a polynomial contains one or more than one terms with a non-zero coefficient.
Example of monomial: 4xy, 89z, 56pqr, -9
Example of binomial: 34x+45y, 9xz+5xy
Example of polynomial: w+ x +y +z, 3r+5t-8z-7p
Let us consider the following cases for division:
Division of monomial by a monomial
This is the simplest case where both the dividend and divisor are monomials. These are factorized and the common factors are cancelled out.
Example 1: Divide 24y3 by 8y
24y3 can be written as 2* 2 * 2 * 3* y * y * y
8y can be written as 2*2*2*y.
So, we need to evaluate (2* 2 * 2 * 3* y * y * y)/(2*2*2*y)
Cancelling the common factors,
We obtain the result as 3 y2 .
Check: 3y2 x 8y =24y3
Example 2. Divide 20mnz by 4 m
Dividend = 4 * 5 * m * n * z
Divisor = 4 * m
Dividing we get, (4* 5* m*n*z)/(4 *m)= 5 n z
Check: 4m* 5nz = 20 mnz
Division of polynomial by a monomial
Example 1: Divide 32(m2np+ mn2p+mnp2) by 4mnp
Dividend= 32(m2np+ mn2p+mnp2) = 4 * 8 *m n p (m +n+ p)
On dividing, (4 * 8 *m n p (m +n+ p)/(4mnp)
= 8(m+ n +p)
Example 2: Divide 36pqr + 12r by 4r
Dividend =36pqr+ 12r =12 r(3pq+1)
So, (36pqr+ 12r ) / 4r can be written as
= 12 r (3pq+1) /4r
= 4r *3 *(3pq+1)/4r
Canceling the common terms,
We get, 3(3pq+1)
Division of polynomial by a polynomial
Sometimes there are some common factors between the dividend and divisor. The strategy here also is to cancel the common factors to obtain the quotient.
Example 1: (12m2 + 24m)/(m+2)
12m2+24 m can be written as 12m(m+2)
So, (12m2 + 24 m)/(m+2) =12m(m+2)/(m+2)= 12m
So the quotient is 12m
Check: 12m × (m+2)= 12m2+24m
We may apply the long division method also to compute the quotient and remainder. The terms of dividend are divided repeatedly by the first term of the divisor.
Example 2: Divide m4-m3+m2 +5 by m+1
This dividend can be rewritten as
Division by m+1 gives
Example 3: Divide m4 – n4 by m2 – n2
By using the identity a2 -b2 = (a-b) (a + b)
We can write m4 – n4 as (m2 – n2)(m2 + n2)
So, cancelling the common terms in dividend and divisor, we get the quotient as m2+n2
Generally an error is a mistake or difference occurred in the calculated value because of the inaccuracy of the calculated number from its true value.
Points to remember to avoid errors:
- Coefficient 1 is usually not written. While adding like terms, do not forget to include it in sum.
e.g. The sum 4x and x+4 is (4x+x+4) i.e. 5x+4 not 4x+4 only.
- While substituting negative values, make use of brackets.
e.g. The product of 5 and (2x+3) i.e. 5(2x+3) is equal to 10x+15 not 10x+9 only.
- When an expression written in a bracket is multiplied by a constant outside the bracket, open the brackets carefully. Make sure that each term of expression has to be multiplied by the constant.
e.g. If it is given that x=-2 then 5x-2 can be written as, 5(-2)-2 which is equal to -12.
- When squaring any monomial, both the contact coefficient and the variable are squared.
e.g. The square of 2x i.e. (2x)2 is equal to 4x2.
- When squaring a binomial, always use the correct formulae.
e.g. The square of (2x+3) i.e. (2x+3)2 is equal to 4x2+9+12x.
- When a polynomial is divided by a monomial then each term of the numerator of the polynomial is divided by the monomial present in the denominator.
This article teaches clearly various methods to do factorization. Also, it discusses the division of algebraic expression with proper examples.
- When a monomial is divided by other monomials, the coefficient of the quotient of two monomials is equal to the quotient of their coefficients.
- When a monomial is divided by other monomials, the variable of the quotient of two monomials is equal to the quotient of the variables of monomials.
- We can check the result by Dividend = Remainder + Divisor * Quotient
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