XXIII.  Special Products

When I first began learning Algebra,  our teacher handed us a sheet of Special Products.  That was 47 years ago, and - somehow I lost it.  To be clear, I did not realize it was special at the time.  I have never found a copy of that sheet.

He told us that if we memorized those (was it 16 products?), we would never have a problem with Algebra.  I never had a problem with Algebra, heck - it even came in handy for Calculus.  There are several websites that talk about special products, but none that I've found (and I've looked) have Mr. Seaton's complete listing.

As with any factoring, always factor out common factors GCF before you begin with anything else.

A blue cone with vertical and horizontal lines drawn on it.

Special Products - SP #1  The Distributive Property

a(b+c) = ab + ac

Hopefully, this looks familiar.  

 SP #2 The Perfect Square Trinomial - Addition

Wow - We used "Trinomial" in the heading!  Trinomial just means we're dealing with three terms.  No big whoop.

(a + b)2 = a2 + 2ab + b2 

SP #3 The Perfect Square Trinomial - Subtraction

(a - b)2 = a2 - 2ab + b2

Let me explain it with an example.  Understand that the a's and b's used in these examples have nothing to do with the a's and b's that are coefficients in the quadratic formula.  a and b can be any combination of variable and constant.  Say we have this expression - 9x2 + 12x + 4  Because I'm actively looking for these special situations, I notice that (3x)2 = 9x2 and 4 is 22!  They are both perfect squares.  In this case a is 3x and b is 2,  so, 2ab = 2*(3x)*(2).  Therefore, according to this special product, 9x2 + 12x + 4 = (3x + 2)2.

SP #4 Factoring The Difference of Two Squares

(a + b)(a - b) = a2 - b2

I would venture to say that this is the most used of all the special products.  A lot of times it comes up, the other way around where we start with the difference of two squares and factor it out to the expression on the left of the equal sign.  Here's an example:  x2 - 9 = (x + 3)(x - 3).  The only hard part here is recognizing that 9 is 32.

SP #5  Factoring a Trinomial

(x + b) (x + c) = x2 + (b + c)x + bc

SP #6 Sum of Variables Cubed

(a + b)3 = a3 + 3s2b + 3ab2 + b3

SP #7 Difference of Variables Cubed

(a - b)3 = a3 - 3a2b + 3ab2 - b3

SP #8 Sum of 2 Cubes

a3 + b3 = (a + b)(a2 - ab + b2)

SP #9 Difference of 2 Cubes

a3 - b3 = (a - b)(a2 + ab + b2)

This is pitiful.  I'm sure there were either 14 or 16 of them.  It appears that only 9 survived.  If you can help me with the missing products, I would appreciate it, check out participation for a rather complex possibility of remuneration for those who assist me in my endeavors.

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Special Products