# Factoring Polynomials Degree 3

A polynomial can be defined as mathematical expression which contains a sum of powers in one or more variables which is
multiplied by coefficients.
The degree of a polynomial is the highest degree of a polynomial’s term whose coefficient is non-zero.

For example
$5x^{2}y^{3}+6x^{2}y+1$
Here there are three terms. First one $5x^{2}y^{3}$ has degree $2+3=5$,
$6x^{2}y$ has degree $2+1=3$ and $1$ has degree 0.
Of this, the highest is 5. So the degree of the polynomial $5x^{2}y^{3}+6x^{2}y+1$ is 5.

Any polynomial whose term has highest degree 3 is called a polynomial of degree 3. It is also called as cubic polynomial.
For example $8x^{3}+6xy+1x$ is a polynomial of degree 3.

Any polynomial of degree 3, in one variable can be generally written as $f(x) = ax^{3}+bx^{2}+cx+d$ where x is the variable and a, b, c and d are the constants.

If a polynomial in the form $a^ {3} ± b^ {3}$ , then it is called perfect cubic polynomial.
It can be factorized using a special formula
$x^ {3} + y^ {3} = (x + y) (x^ {2}-x y + y^ {2})$
$x^ {3} - y^ {3} = (x - y) (x^ {2} +x y + y^ {2})$

Factoring a polynomial is a process by which a polynomial is written as the product of two or more simpler polynomial. It must be noted that all polynomial cannot be written as product of factors. Such polynomials are called prime polynomials.
For example $5u^ {3} + 3w^{2})$

To factorize, the first thing is to check if there exist a greatest common factor (GCF), generally called, common factor, in the polynomial.
For example $7x^{3} – 14x^{2}$
Here there are two terms $7x^{3}$and $14x^{2}$
$7x^{3}=1*7*x*x*x$
$14x^{2}= 1*2*7*x*x$
Here GCF=$1*7*x*x=7x^{2}$
Write this GCF outside the parenthesis
Here $7x^(2)( )$
Divide each of the term by the GCD obtained.
Here, $\frac{7x^{3}}{7x^{2}}= x$ and $\frac{14x^{2}}{7x^{2}}=2$
Plug them inside the parentheses
7x2(x-2)
So, 7x3 – 14x2 = 7x2(x-2)

## Factoring polynomials by grouping

In certain cases some polynomials of degree 3 may have about four terms and there exists no common term (GCF) for all the terms.
In such situation, factoring is possible by the process of grouping.
Here the main procedure is to factor the polynomial by setting the terms into two group and then factoring the common term (GCF) out of each group.

Lets consider an example, and go through the steps that has to be followed when factoring the polynomial by grouping.
$4x^{3}+6x^{2}-8x-12$
Separate the polynomial terms into two sets, by grouping the first two terms together and then the last two terms together.
$(4x^{3}+6x^{2})+(-8x-12)$
Identify a common factor in each of the separate groups.
Lets take $(4x^{3}+6x^{2})$
$4x^ {3} =1*2*2*x*x*x$
$6x^ {2} =1*2*3*x*x$
So common factor = $1*2*x*x=2x^ {2}$
Lets take $(-8x-12)$
$-8x = -1*2*2*2*x$
$-12 =-1*2*2*3$
So common factor = $-1*2*2= -4$

$4x^ {3} +6x^ {2}-8x-12= 2x^ {2} (2x–3) + (-4) (2x - 3)$
= $2x^ {2}(2x– 3) - 4(2x - 3)$

Identify the common term in each group
In $2x^ {2} (2x– 3)$and -4(2x -3), the common term is (2x-3)
Factor it out using reverse of distribution property
(2x-3)(2x^ {2} – 4)
So, $(4x^ {3} +6x^ {2}) + (-8x-12) = (2x-3) (2x^ {2} – 4)$
Now, in this problem, $(2x^ {2} – 4)$ can be factored again, as there is a $2$ common to them.
So, $(2x^ {2} – 4) =2(x^ {2}-2)$
So, $4x^ {3} +6x^ {2}-8x-12= (2x-3)2(x^{2}-2)$
= $2(2x-3)(x^ {2}-2)$