Factors of Algebraic Expressions - Part I

We know what it means to factorize a number. When we factorize a number, we split it into a series of smallest indivisible numbers, which when multiplied together give the original number. To find the factors of algebraic expressions, a similar method is followed.

What we will study in this chapter :

1. Factors of algebraic monomials
2. Division of monomials
3. Factors of binomials
4. Factors of a polynomial by grouping the terms

1. Factors of algebraic monomials  
Try and factorize monomials the way we factorize natural numbers.

Example 1 :  Factorize 5a

The monomial  5a can be factorized as a product of two monomials 5 and a.

Thus  5a   =  5   *   a

The two monomials 5 and a are factors of 5a.

Example 2 :  Factorize 3a³b².

The monomial 3a³b²  can be written as

 3a³b²   =  3 a³b¹  *  b

 3 a³b¹ can be factorized further.

 3 a³b¹ = 3 a³ *  b,  thus 3a³b²   =  = 3 a³ *  b *  b

 3 a³  can be factorized further.

 3 a³ =  3 a² *  a, thus 3a³b²   =  = 3 a² *  a *  b *  b

 3 a²  can be factorized further.

 3 a² =  3 a *  a, thus 3a³b²   =  = 3 a *  a *  a *  b *  b

Lastly 3a can be factorized further.

3a = 3 * a, thus 3a³b²   =  = 3 * a *  a *  a *  b *  b

The factorization is complete.

While factorizing a monomial, the following has to be remembered :

• Obtain prime factors of the coefficient. In the above example, the coefficient is 3, which is a prime number. If it was say 6, the we would have had to split 6 again to its factors 6 = 2 * 3.

• Factorize the variables separately.

• Each index of the variable should be reduced, till the index  is 1.

• All the factors of the coefficient and the variables should be written in the form of a multiplication.

Example 3 :  factorize 9a³b².

Factors of 9  : 3 * 3

Factors of a³  :  a  *  a  *  a

Factors of b²:  b  * b

  9a³b²  =  3 * 3   * a  *  a  *  a  * b  * b

Example 4 : Find the common factor of 25a²b, 15 a²bc and 5 a³b².

The G.C.D. of 25, 15, 5 is  :  5.

The lowest power of the common variable a is  :  a².
The lowest power of the common variable b is  :  b.
Variable c has no common factor.

Thus the lowest common factor of the given monomials is 5 * a² * b  = 5 a²b.

2. Division of monomials  
For dividing one monomial with another, factorize both the numerator and the denominator. The remaining coefficients and variables will give you the final answer.

Example 1 :  Divide 5ab by a

   5ab
   = 5b
    a

Thus 5ab    a  =  5b

Example 2 :  Divide 24x²yz by 12xy

   24x²yz                        2 * 3 * 4 *   x  *  x  *  y  *  z
            =           =  2xz
    12xy                                     3 * 4 *  x * y

Thus   24x²yz      12xy  = 2xz

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