Algebra 1 Binomial Expansion
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A Triangle
Binomial Expansion

 

 There are some factors that show up all the time.

 It is a good idea to get to know these.
 They are ...
 
 The Perfect Squares:
 Using "a" and "b" to stand for any old numbers we might have,
 the perfect squares are:
 

 (aX + b)2 = a2X2 + 2abX + b2

(aX - b)2 = a2X2 - 2abX + b2

 
 When these two are multiplied out like this 
 they are almost the same.
 
 The only difference is that the one that starts with a minus
 has a minus on the middle term.
 
 So using real numbers and multiplying these out 
 we would get ...
 
 Examples:
 

(X + 3)2  =  12X2 + 2x1x3xX + 32  =  X2 + 6X + 9
 
  (3X + 5)2  =  32X2 + 2x3x5xX + 52  =  9X2 + 30X + 25
 
  (X2 - .5)2  =  .52X4 - 2x1x.5xX2 + .52  =  .25X4 - X2 + .25
 
 The Perfect Cubes:
 Using "a" and "b" to stand for any old numbers we might have,
 the perfect cubes are:
 

 (aX + b)3 = a3X3 + 3a2bX2 + 3ab2X + b3

(aX + b)3 = a3X3 - 3a2bX2 + 3ab2X - b3

 
 When these two are multiplied out,
 the only difference between them is also in the signs.
 The signs on (aX + b) 3 are all positive
 The signs on (aX - b) 3 alternate: positive - negative - positive - negative
 
 So using real numbers and multiplying these out 
 we would get ...
 
 Examples:
 
  (X + 3)3  =  13X3 + 3x12x3xX2 + 3x1x32xX + 33  =  X3 + 9X2 + 27X + 9
 
  (X - p)3  =  13X3 - 3x12xpxX2 + 3x1xp2xX - p3  =  X3 + 3pX2 + p2X + p3
 
 We could keep on going with this stuff forever.
 We could do (aX + b) 4, (aX - b)7 and on and on,
 but there is an easier way ...
 
 It's called Pascal's triangle.
 
 It starts like this.

 1          1

 
 For our purposes here,
 these stand for the two terms in something you 
 would be multiplying times itself.
 
 Since this is the start,
 it shouldn't be too surprising that this stands for:
 

 (a + b)1   which is just a + b

 
 The next step builds on the start above (the 1   1).
 It looks like this ...
 

 
 The following levels continue the pattern.
 

 
 OK, this looks really cute.
 BUT WHAT GOOD IS IT?
 
 Here's the deal ...
 

 
 See what happens?
 The coefficients are the numbers from the triangle.
 Check out what the exponents on X are doing.
 It's a countdown ...
 
 We could have a coefficient on X.
 
 Or a number other than 1.
 
 To deal with that, 
 here's the first 4 levels of the triangle
 with "a" and "b" standing for any old numbers we might have.
 

 
 Look what happens.
 The numbers in bold are the Pascal's triangle numbers.
 The exponents on X go down 1 every term as we go from left to right.
 The exponents on "a" go down 1 every term as we go from left to right.
 The exponents on "b" go up 1 every term as we go from left to right.
 
 This stuff works real well, but it takes up a lot of space.
 If I wanted to write out a Pascal's triangle
 for exponents up to, say, 50,
 It would take up an amazing amount of space.
 Math types try to save space anywhere they can.
 
 They came up with a shorthand way to tell someone
 what any level of the triangle would look like.
 
 It goes like this ...
 
 Using "a" "b" and "n" to stand for 
 any coefficient or exponent we might have ...
 
 (aX + b)n = anXn - nan-1bXn-1 ... nabn-1X + bn
 
 This is the start of something called the binomial theorem.
 

   copyright 2005 Bruce Kirkpatrick

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