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Factoring Expressions

Expressions are the ways of saying something. When a statement is expressed in the language of mathematics, it is called a mathematical expression. Basically we learn expressions in algebra, the most useful branch of mathematics. When symbols, numbers and operators are written together in order to represent the value of something, they form an expression. The numbers are known as constants. In algebra, when the number are unknown, the letters or symbols are used. These letters are called variables. The expressions, better known as algebraic expressions, are combinations of variables and constants by the use of operators, such as +, -, x.

For example -

(i)
 $6\ \times\ 7$ is an expression having two constant multiplied together.

(ii)
 $9a$ is an expression in which a constant and a variable are multiplied together. Here, the multiplication sign is left, since its presence is              obvious. So, rather writing 9 x a, we write 9a.

(iii)
 $6x\ +\ y$ is a linear expression.

(iv)
 $4^{2}\ -\ 7x\ +\ 3$ is a quadratic expression.

(v)
 $(x\ +\ 1)\ (y\ -\ 6)$ is also an expression.

Thus, we can say that expression is a much broader term in mathematics. We can say that an expression is a string of letters and numbers joined together by the help of algebraic operations, plus, minus and multiply. In this article, we are going to focus on factorization of algebraic expressions.

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Factoring Expressions Definition

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The factor of a number means breaking it up into smaller (sometimes equal too) numbers which if multiplied together give original number.

For example - One of the factors of 6 are 3 and 2.

There may be more than two factors as well, such as 8 can be expressed as $4\ \times\ 2$ or $2\ \times\ 2\ \times\ 2$.

When we talk about algebraic expressions, factorization is the process of finding factors of an expression. So, the factorization or factoring is known as decomposition of an expression in the form of a product of other expressions which on multiplication, give the original expression. The expressions obtained after factorizing an expression, are known as factors. For example - The expression $x^{2}\ -\ 4$ has factors $(x\ -\ 2)(x\ +\ 2)$, where $x\ -\ 2$ and $x\ +\ 2$ are individually called factors of $x^{2}\ -\ 4$.

The factors leave no remainder in division. More elaborately, when an expression is divided by any of its factors, it leaves zero as remainder, i.e., an expression is completely divided by its factors.

How To Factor Algebraic Expressions

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There are many different ways of factoring expressions. The most useful are listed below.

(1)
Highest Common Factor
     The highest common factor is factored out of the expression.

(2)
 Factoring Expressions by Grouping
     When there is no HCF common to all the terms of an expression, the groups of two or more terms can be formed and            each group can factorized      further.

(3)
 Using Quadratic Formula
     The quadratic equations can be factored using quadratic formula which will be discussed in detail down in this page.

(4) Difference of Square Formula

     When there is difference of squares, the following formula is used -
     $a^{2}\ -\ b^{2}\ =\ (a + b)(a - b)$

(5)
 Sum or Difference of Cubes
     When we have sum of two cubes or difference of two cubes in the expression, we may factorize it using following                  formulae - $a^{3} + b^{3} = (a + b) (a^{2} + b^{2} - ab)$
                     $a^{3} + b^{3} = (a - b) (a^{2} + b^{2} + ab)$
     
(6) Making Perfect Squares
     A quadratic expression can also be factorized by completing the square using square of sum and square of difference            formulae - $a^{2} + b^{2} + 2ab = (a+b)^{2}$
                     $a^{2} + b^{2} - 2ab = (a-b)^{2}$
     

Factoring Quadratic Expressions

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In order to factorize quadratic equation, one needs to follow the following procedure.

(1)
 If there is any common factor among all the terms, first factor it out.
     $2x^{2}\ -\ 4x\ -\ 30$
  = $2(x^{2}\ -\ 2x\ -\ 15)$

(2) Keep common factor (if any) outside the bracket and factorize the quadratic expression using the product-sum method which is discussed              below.

Suppose we have expression of the form $at^{2}\ +\ bt\ +\ c$, then we have to find two numbers whose product is equal to ac and sum is equal to b. If $b_{1}$ and $b_{2}$ are numbers, then we must have $b_{1} \times\ b_{2}\ =\ ac$ and $b_{1}\ +\ b_{2}\ =\ b$

Substitute this sum in place of b and then factorize by grouping.

In above example -

We may have -5 and 3, since (-5) x 3 = -15 (ac) and -5 + 3 = -2 (b)
   $2(x^{2}\ -\ 2x\ -\ 15)$
= $2[x^{2}\ +\ (- 5\ +\ 3)\ x\ -\ 15]$
= $2[x^{2}\ -\ 5x\ +\ 3x\ -\ 15]$
= $2[x(x\ -\ 5)\ +\ 3(x\ -\ 5)]$
= $2\ (x\ -\ 5)\ (x\ +\ 3)$

Factoring Rational Expressions

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A rational expression is defined as a fraction having numerator both denominator are represented in terms of polynomials. Few examples of rational expressions are :

$\frac{4x}{x-1}$, $\frac{(m-3)(4n - 5)}{m(n -1)}$, $\frac{t^{2}-1}{t^{3}- 27}$ etc.

In order to factorize rational expressions, we are required to follow the steps mentioned below.
Step 1 : 

If there is any number common to the terms in numerator, then factor it out. Similar procedure should be applied to denominator as well.

For example -

$\frac{t^{2}-9}{5t+15}$

There is no common factor in numerator, while 5 can be factored out from the denominator.

= $\frac{t^{2}-9}{5(t+3)}$
Step 2 : 

Now, factorize the numerator and denominator by any method of factorizing algebraic expression.

   $\frac{t^{2}-9}{5(t+3)}$

= $\frac{t^{2}-3^{2}}{5(t+3)}$

= $\frac{(t+3)(t-3)}{5(t+3)}$
Step 3 : 

Cancel out common terms in numerator and denominator and solve until getting simplest form.

$\frac{(t+3)(t-3)}{5(t+3)}$

= $\frac{t-3}{5}$

Here, t + 3 cancels out.

Factoring Expressions Examples

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Let us have a look at some solved examples of factoring algebraic expressions.
Example 1 : 

Factorize $5a^{2}\ -\ 500$

Solution :
 

$5a^{2}\ -\ 500$

= $5(a^{2}\ -\ 100)$

= $5(a^{2}\ -\ 10^{2})$

Using the identity $a^{2}\ -\ b^{2}\ =\ (a\ +\ b)\ (a\ -\ b)$, we get

= $5(a\ +\ 10)\ (a\ -\ 10)$
Example 2 : 

Factorize the following expression : $x^{3} + 64y^{3}$

Solution :
 

$x^{3} + 64y^{3}$

= $x^{3} + (4y)^{3}$

Using identity $a^{3} + b^{3} = (a + b) (a^{2} + b^{2} - ab)$, we get

= $(x + 4y)[x^{2} + (4y)^{2} - x.(4y)]$

= $(x + 4y)(x^{2} + 16y^{2} - 4xy)$
Example 3 : 

Find the factors of expression $3x^{2}\ +\ 4x\ -\ 4$.

Solution : 

   $3x^{2}\ +\ 4x\ -\ 4$

= $3x^{2}\ +\ 6x\ -\ 2x\ -\ 4$

= $3x(x\ +\ 2)\ -\ 2(x\ +\ 2)$

= $(x\ +\ 2)\ (3x\ -\ 2)$
Example 4 : 

Factorize the following rational expression $\frac{t^{2}\ +\ 6t\ +\ 5}{t^{2}\ -\ t\ -\ 2}$.

Solution :
 

   $\frac{t^{2}\ +\ 6t\ +\ 5}{t^{2}\ -\ t\ - 2}$

= $\frac{t^{2}\ +\ 5t\ +\ t\ +\ 5}{t^{2}\ -\ 2t\ +\ t\ -\ 2}$

= $\frac{(t\ +\ 5)\ (t\ +\ 1)}{(t\ -\ 2)(t\ +\ 1)}$

= $\frac{(t\ +\ 5)}{(t\ -\ 2)}$
Example 5: 

Factorize $\frac{5x^{2}\ +\ 15x}{x^{2}\ -\ 8x\ -\ 33}$.

Solution:
 

   $\frac{5x^{2}\ +\ 15x}{x^{2}\ -\ 8x\ -\ 33}$

= $\frac{5x(x\ +\ 3)}{x^{2}\ -\ 11x\ +\ 3x\ -\ 33}$

= $\frac{5x(x\ +\ 3)}{(x\ -\ 11)(x\ +\ 3)}$

= $\frac{5x}{(x\ -\ 11)}$ 
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