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.

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 3x

^{2}y – 9xy + 12xy

^{2}.

**Solution:**

**Step 1:**

Given 3x

^{2}y – 9xy + 12xy

^{2 },

^{ }is a trinomial

Algebraic expression has two variables, having constants and the terms are connected by addition/subtraction operators.

Step 2:

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:

Step 3:

The GCF of the variable ‘x’ in x

^{2}y, xy and xy

^{2}is ‘x’, because only ‘x’ can divide all the terms 3x

^{2}y – 9xy + 12xy

^{2}.

The GCF of the variable ‘y’ in x

^{2}y, xy and xy

^{2}is ‘y’, because only ‘y’ can divide all the terms 3x

^{2}y – 9xy + 12xy

^{2}.

Step 4:

Step 4:

The given expression can therefore be written by factoring as,

=> 3x

^{2}y – 9xy + 12xy

^{2}= 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).

=> 3x

^{2}y – 9xy + 12xy

^{2}= 3xy(x + 4y – 3).

**Question 2:**Factor the trinomial x

^{2}- 15x + 36.

**Solution:**

**Step 1:**

Given x

^{2}- 15 x + 36

This has no common monomial factor

Step 2:

Step 2:

Solve for x,

=> x

^{2}- 15 x + 36 = x

^{2}- 12 x - 3x + 36

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

= (x - 3)(x - 12)

=> x

^{2}- 10 x + 36 = (x - 3)(x - 12).

**answer**

**Question 3:**Factor: 1 - 49y

^{4}

**Solution:**

Given 1 - 49y

Factorized, 1 - 49y

= (1 + 7y

[ a

The square root of 1 is 1, and the square root of 49y

The sum (1 + 7y

=> 1 - 49y

^{4}Factorized, 1 - 49y

^{4}=> 1 - 49y^{4}= 1^{2}- (7y^{2})^{2}= (1 + 7y

^{2})(1 - 7y^{2})[ a

^{2}- b^{2}= (a + b)(a - b) ]The square root of 1 is 1, and the square root of 49y

^{4}is 7y^{2}.The sum (1 + 7y

^{2}) and difference (1 - 7y^{2}) of the square root are the factors.=> 1 - 49y

^{4}= (1 + 7y^{2})(1 - 7y^{2})**answer****Question 4:**Factoring algebraic expression: 12x(x + 2)(x + 3)

^{2}+ 21(x - 4)(x + 3)

**Solution:**

Given 12x(x + 2)(x + 3)

Find the common factors of given algebraic expression

=> 12x(x + 2)(x + 3)

= 3 * 4 * x (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)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 - 2^{2})(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)}.