Explore: Factoring Polynomials Completely Solutions

To factor completely means that you cannot factor the expression any further. 
To factor completely, combine multiple methods of factoring in the following order:

1. Find the greatest common monomial factor (GCF) (other than 1). 
2. Factor by grouping (when given 4 terms). 
3. Analyze the sign patterns.
4. Factor special products (ex. difference of two squares, perfect square trinomials).
5. Factor using your preferred factoring method (ex. ac-grouping, modeling, or mental math).

Some expressions cannot be factored . When this occurs, answer “cannot be factored .”
To check if you have factored an expression correctly, multiply the product of terms back together to see if the given expression results.

Note

An in-depth exploration of factoring can be found in Algebra 1.

Example 7

Factor completely.
$8 x^{2} - 6 x - 44$

Implement

$2 (4 x^{2} - 3 x - 22) 2 (x + 2) (4 x - 11)$

Explain

Factor out the GCF
Use sign patterns to factor

Example 8

Factor completely.
$4 x^{4} - 36 x^{2} - x^{2} y^{2} + 9 y^{2}$

Implement

$(4 x^{4} - 36 x^{2}) + (- x^{2} y^{2} + 9 y^{2}) 4 x^{2} (x^{2} - 9) - y^{2} (x^{2} - 9) (x^{2} - 9) (4 x^{2} - y^{2}) (x - 3) (x + 3) (2 x + y) (2 x - y)$

Explain

Group terms
Factor out the GCF from each group of terms
Regroup terms
Factor difference of two squares