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Factoring Algebraic Expressions

An algebraic expression is a statement containing all possible combinations of variables, constants and operators. Factoring is a technique in driving the factors of a algebraic expression. An algebraic expression may be subjected into different operations and can be simplified, translated and evaluated. Algebraic expressions can be factored by applying common monomial factoring using greatest common factor, applying factoring bionomials like the difference of two square, sum and difference of two cubes, of two prime powers and of two composite powers, applying factoring trinomials.

How to Factor Algebraic Expressions

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An equation is nothing but a statement which says two expressions are equal. We will illustrate about this later in an example problem. Algebraic expressions also in the form of statement which states that two expressions are equal. The concept of factoring algebraic expression is extremely important for factoring the equations. Before solving a equation we need to know how to perform the basic operations on algebraic expressions within the equation. Factoring is a art that one can learn with experience and the help of useful techniques. Lets see with the help of some examples how to factor a algebraic expression.

Solved Examples

Question 1: Simplify the expression by finding the GCF 3x2y – 9xy + 12xy2.
Solution:
Step 1:
Given 3x2y – 9xy + 12xy, is a trinomial
Algebraic expression has two variables, having constants and the terms are connected by addition/subtraction operators.

Step 2:

Finding the GCF of the constants and the variables one by one.

The GCF of the constants in 3, 9 and 12 is 3, because only 3 can divide all the numbers 3, 9 and 12.

=> GCF(3, 9, 12) = 3

Step 3:

The GCF of the variable ‘x’ in x2y, xy and xy2 is ‘x’, because only ‘x’ can divide all the terms 3x2y – 9xy + 12xy2.

The GCF of the variable ‘y’ in x2y, xy and xy2 is ‘y’, because only ‘y’ can divide all the terms 3x2y – 9xy + 12xy2.

Step 4:

The given expression can therefore be written by factoring as,

=> 3x2y – 9xy + 12xy2 = 3xy*(x) – 3xy*(3) + 3xy*(4y).

Now by using distributive property, it can be simplified as,

=> 3xy(x – 3 + 4y)

or  3xy(x + 4y – 3).

=> 3x2y – 9xy + 12xy2 = 3xy(x + 4y – 3).
 

Question 2: Factor the trinomial  x2 - 15x + 36.
Solution:
Step 1:
Given x2 - 15 x + 36
This has no common monomial factor

Step 2:

Solve for x,

=> x2 - 15 x + 36 = x2 - 12 x - 3x + 36

= x(x - 12) - 3(x - 12)

= (x - 3)(x - 12)

=> x2 - 10 x + 36 = (x - 3)(x - 12). answer
 

Question 3: Factor: 1 - 49y4
Solution:
Given 1 - 49y4
Factorized, 1 - 49y4

 => 1 - 49y4 = 12 - (7y2 )2

= (1 + 7y2 )(1 - 7y2 )

[ a2 - b2 = (a + b)(a - b) ]

The square root of 1 is 1, and the square root of 49y4 is 7y2.

The sum (1 + 7y2) and difference (1 - 7y2) of the square root are the factors.

=> 1 - 49y4 = (1 + 7y2 )(1 - 7y2 ) answer
 

Question 4: Factoring algebraic expression: 12x(x + 2)(x + 3)2  + 21(x - 4)(x + 3)
Solution:
Given 12x(x + 2)(x + 3)2  + 21(x - 4)(x + 3)

Find the common factors of given algebraic expression

=> 12x(x + 2)(x + 3)2  + 21(x - 4)(x + 3) = 3 * 4 * x (x + 2)(x + 3)2  + 3 * 7(x - 22)(x + 3)

= 3 * 4 * x (x + 2)(x + 3)2  + 3 * 7(x - 2)(x + 2)(x + 3)

Algebraic expression has as a common factor = 3(x + 2)(x + 3)

= 3(x + 2)(x + 3){4x(x + 3) + 7(x - 2)}

=> 12x(x + 2)(x + 3)2  + 21(x - 4)(x + 3) =  3(x + 2)(x + 3){4x(x + 3) + 7(x - 2)}.