Dynamic Distributions of Binomials
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What are binomials? A binomial is a quantity sum of two terms: (a+b), (2+3)4, (1+x)2, etc. To expand these binomials means to remove the ( )’s. First, we recall, any factor raised to the zero index or power is equal to one (1), ie - (x+y)0 = 1, next, any factor raised to the exponent, index, power all meaning the same (1), ie (x+y)1 or (x+y) = x+y, (7+3)1 = 7+3. Let’s try some examples 70=1, (7+0)1 = 7+0, (w-y)0 = 1, (w-y)1 or (w-y) = w-y, (n-1)0 = 1, on and on. Moving on, with a quantity binomial raised to some index or power: (a+b)n. Let n=2, then (a+b)2 may be expanded by stacked multiplication: This is a process we learned in H.S. algebra. Another is : Horizontal Foil (a+b)2 = (a+b)(a+b) the other is: Binomial theorem or ... Now let’s try something NEW! ‘SALCIDO’S VERTICAL FOIL’ Table Foil Terms Term # 12 ÷ 1 = 2 1 a2 b0 1 21 ÷ 2 = 1 2 a1 b1 2 1 a0 b2 3 Notice the index of (a+b)2 is 2. which, also tells us there are three terms in the expansion. Here’s how you do it. Set up the basic ‘Salcidos Vertical Foil’ Table (later on other label headings may be added: Foil Terms Term #’ then: Fill in the term # Col. ‘Foil Terms Term #’ 1 2 3 then: Fill in the Terms col. - Foil Terms Term # 12 ÷ 1 = 2 1 a2 b0 1 21 ÷ 2 = 1 2 a1 b1 2 1 a0 b2 3 a2 + 2ab + b2 Then do the foil operations under the foil column. Let’s try: (a+b)4 Foil Terms Term # 14 ÷ 1 = 4 1 a4 b0 1 43 ÷ 2 = 6 4 a3 b1 2 62 ÷ 3 = 4 6 a2 b2 3 41 ÷ 4 = 1 4 a1 b3 4 1 a0 b4 5 a4 + 4a3b + 6a2b2 + 4ab3 + b4 A little more difficult one: (a+b)10
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About the Book
As you read in the book, the details will be brought about as they came to me throughout the 50 years or so. I guess the catalyst of this book was the one statement made to the class (HS) around 1962 or 1963 – “when a binomial is expanded to its power, there will be one more number of terms than the exponent number”. This book “Dynamic Distributions of Binomials” will show you what I found to be a useful paradigm.
About the Author
I was born in Globe, AZ on April 17, 1946. I graduated from Miami High School, Miami Arizona. I spent four great years in the Air Force (1965 – 1969). Math has been with me and influenced me to be involved with it all my life as far back as I can remember. For some reason or other, binomial expansions and foiling binomials, caught my interest early in high school. Mr. Danny Kinnard, math teacher at Miami HS was my hero. He could manipulate the Binomial theorem like magic. On and off from the early sixties ‘til a few years ago, I’ll say 2007 did I start really putting a lot of time into this book. As you read in the book, the details will be brought about as they came to me throughout the 50 years or so. I guess the catalyst of this book was the one statement made to the class (HS) around 1962 or 1963 – “when a binomial is expanded to its power, there will be one more number of terms than the exponent number”. This book “Dynamic Distributions of Binomials” will show you what I found to be a useful paradigm.