# 6.4 General Strategy for Factoring Polynomials

Topics covered in this section are:

1. Recognize and use the appropriate method to factor a polynomial completely

## 6.4.1 Recognize and Use the Appropriate Method to Factor a Polynomial Completely

### HOW TO: Use a general strategy for factoring polynomials.

1. Is there a greatest common factor?
Factor it out.
2. Is the polynomial a binomial, trinomial, or are there more than three terms?
If it is a binomial:
• Is it a sum?
Of squares? Sums of squares do not factor.
Of cubes? Use the sum of cubes pattern.
• Is it a difference?
Of squares? Factor as the product of conjugates.
Of cubes? Use the difference of cubes pattern.If it is a trinomial:
• Is it of the form $x^{2}+bx+c$? Undo FOIL.
• Is it of the form $ax^{2}+bx+c$?
If $a$ and $c$ are squares, check if it fits the trinomial square pattern.
Use the trial and error or “ac” method.
If it has more than three terms:
• Use the grouping method.
3. Check.
Is it factored completely?
Do the factors multiply back to the original polynomial?

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

#### Example 1

Factor completely: $7x^{3}-21x^{2}-70x$.

Solution

Be careful when you are asked to factor a binomial as there are several options!

#### Example 2

Factor completely: $24y^{2}-150$.

Solution

The next example can be factored using several methods. recognizing the trinomial squares pattern will make your work easier.

#### Example 3

Factor completely: $4a^{2}-12ab+9b^{2}$.

Solution

Remember, sums of squares do not factor, but sums of cubes do!

#### Example 4

Factor completely: $12x^{3}y^{2}+75xy^{2}$.

Solution

When using the sum or difference of cubes pattern, be careful with the signs.

#### Example 5

Factor completely: $24x^{3}+81y^{3}$.

Solution

#### Example 6

Factor completely: $3x^{5}y-48xy$.

Solution

#### Example 7

Factor completely: $4x^{2}+8bx-4ax-8ab$.

Solution

Taking out the complete GCF in the first step will always make your work easier.

#### Example 8

Factor completely: $40x^{2}+44xy-24y$.

Solution

When we have factored a polynomial with four terms, most often we separated it into two groups of two terms. Remember that we can also separate it into a trinomial and then one term.

#### Example 9

Factor completely: $9x^{2}-12xy+4y^{2}-49$.

Solution

$(3x-2y-7)(3x-2y+7)$
$9x^{2}-6xy+21x-6xy+4y^{2}-14y-21x+14y-49$
$9x^{2}-12xy+4y^{2}-49 \ \checkmark$