2.2. Advanced Factoring http://www.ck12.org
2.2 Advanced Factoring
Here you will be exposed to a variety of factoring techniques for special situations. Additionally, you will see
alternatives to trial and error for factoring.
The difference of perfect squares can be generalized as a factoring technique. By extension, any difference
between terms that are raised to an even power likea^6 −b^6 can be factored using the difference of perfect squares
technique. This is because even powers can always be written as perfect squares:a^6 −b^6 = (a^3 )^2 −(b^3 )^2.
What about the sum or difference of terms with matching odd powers? How can those be factored?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/60702http://www.youtube.com/watch?v=55wm2c1xkp0 James Sousa: Factoring: Trinomials using Trial and Error and
Grouping
Guidance
Factoring a trinomial of the formax^2 +bx+cis much more difficult whena 6 =1. In Examples A and B, you
will see how the following expression can be factored using educated guessing and checking and the quadratic
formula. Additionally, you will see an algorithm (a step by step procedure) for factoring these types of polynomials
without guessing. The proof of the algorithm is beyond the scope of this book, but is a reliable technique for getting
a handle on tricky factoring questions of the form:
6 x^2 − 13 x− 28
When you compare the computational difficulty of the three methods, you will see that the algorithm described in
Example A is the most efficient.
A second type of advanced factoring technique is factoring by grouping. Suppose you start with an expression
already in factored form:
( 4 x+y)( 3 x+z) = 12 x^2 + 4 xz+ 3 xy+yz
Usually when you multiply the factored form of a polynomial, two terms can be combined because they are like
terms. In this case, there are no like terms that can be combined. In Example C, you will see how to factor by
grouping.
The last method of advanced factoring is the sum or difference of terms with matching odd powers. The pattern is:
a^3 +b^3 = (a+b)(a^2 −ab+b^2 )
a^3 −b^3 = (a−b)(a^2 +ab+b^2 )
This method is shown in the guided practice and the pattern is fully explored in the exercises.