How to factor polynomials

The methods that you will use to factor any polynomial depend on how many terms the polynomial has.

Any Polynomial

The first step, no matter what you are factoring, is always to factor out the Greatest Common Factor, commonly referred to as the GCF.
For example:

${\displaystyle 2a^{2}b^{3}+4a^{4}b^{4}c^{3}-2a^{3}b^{3}=2a^{2}b^{3}(1+2a^{2}bc^{3}-a),\,\!}$ 

or

${\displaystyle 3x^{3}-6x+9x^{4}=3x(x^{2}-2+3x^{3}),\,\!}$ 

or

${\displaystyle 4x(x+2)+3x^{2}(x+2)=(x+2)(4x+3x^{2}),\,\!}$ 

Binomial--2 Terms

Again, the first step is to factor out the GCF. If there is no GCF, then there are only 3 possibilities: Difference of Squares, Sum of Cubes, or Difference of Cubes.

Difference of Squares:

${\displaystyle x^{2}-y^{2}=(x+y)(x-y),\,\!}$ 

For example:

${\displaystyle y^{2}-9=(y+3)(y-3),\,\!}$ 

or

${\displaystyle 16a^{2}-49b^{2}=(4a+7b)(4a-7b).\,\!}$ 

Sum of Cubes:

${\displaystyle x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2}),\,\!}$ 

For example:

${\displaystyle z^{3}+27=(z+3)(z^{2}-3z+9),\,\!}$ 

or

${\displaystyle 8x^{3}+125=(2x)^{3}+(5)^{3}=(2x+5)[(2x)^{2}-(5)(2x)+(5)^{2}]=(2x+5)(4x^{2}-10x+5).\,\!}$ 

Difference of Cubes:

${\displaystyle x^{3}-y^{3}=(x-y)(x^{2}+xy+y^{2}),\,\!}$ 

For example:

${\displaystyle z^{3}-27=(z-3)(z^{2}+3z+9),\,\!}$ 

or

${\displaystyle 8x^{3}-125=(2x)^{3}-(5)^{3}=(2x-5)[(2x)^{2}+(5)(2x)+(5)^{2}]=(2x-5)(4x^{2}+10x+5).\,\!}$ 

Trinomial--3 Terms

There are three possibilities for factoring a trinomial depending on which type of trinomial it is.

Monic Trinomials

A monic trinomial has a 1 as the leading coefficient.
${\displaystyle x^{2}+bx+c=(x+d)(x+e),}$ 

where

${\displaystyle ed=c}$ 

and

${\displaystyle d+e=b}$ 

For example:
${\displaystyle x^{2}-x-6=(x-3)(x+2)}$  because ${\displaystyle (-3)(2)=-6}$  and ${\displaystyle -3+2=-1}$ 
${\displaystyle x^{2}-11x+24=(x-3)(x-8)}$  because ${\displaystyle (-3)(-8)=24}$  and ${\displaystyle -3+-8=-11}$ 

Non-Monic Trinomials

A non-monic trinomial has a constant other than 1 as the leading coefficient.
${\displaystyle ax^{2}+bx+c=(mx+p)(nx+q)}$ 
where ${\displaystyle mn=a}$ , ${\displaystyle pq=c}$ , and ${\displaystyle mq+pn=b}$ 
Many times students are taught that to factor a non-monic trinomials, they must guess different combinations of m,n,p,and q and then FOIL the factors to see if they had guessed correctly. There is a method of factoring that, while not often taught, will work.
Example: Factor ${\displaystyle 6x^{2}+7x-3}$ 
Step 1 Multiply a and c. (Multiply the number in front of ${\displaystyle x^{2}}$  and the constant)
Multipy 6 and -3, ${\displaystyle 6(-3)=-18}$  Step 2 Find factors of ac.
Find factors of -18: -1(18), 1(-18), -2(9), 2(-9), -3(6), and 3(-6).

Step 3 Decide which factors of ac that when added together will give b.
The combination of -2 and 9 is the one needed since -2+9=7.

Step 4 Rewrite the middle term of bx using the factors found in step 3. Instead of ${\displaystyle 7x}$ , write ${\displaystyle 6x^{2}-2x+9x-18}$ 
Step 5 Factor by grouping.
${\displaystyle (6x^{2}-2x)+(9x-3)}$ 
${\displaystyle 2x(3x-1)+3(3x-1)}$  ${\displaystyle (3x-1)(2x+3)}$ 

Therefore ${\displaystyle 6x^{2}+7x-3=(3x-1)(2x+3)}$ 

Perfect Square Trinomials

Perfect square trinomials are of the form ${\displaystyle a^{2}+2ab+c^{2}}$  or ${\displaystyle a^{2}-2ab+c^{2}}$ 
${\displaystyle a^{2}+2ab+c^{2}=(a+c)^{2}}$  and
${\displaystyle a^{2}-2ab+c^{2}=(a-c)^{2}}$ 
For example: ${\displaystyle 4h^{2}-20h+25=(2h)^{2}-2(2h)(5)+(5)^{2}=(2h-5)^{2}}$ 

References

External links

• example.com