Algebra 2 Special Zeros of an Equation
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Just a Touch
Special Zeros of an Equation

 

 So we know that if we have an equation 

 that factors to look like this:

 Y = (X - 2)(X + 3)(X - 1)

 
 The graph will cross the X axis (where Y = 0)
 when X = 2, X = -3, and X = 1.
 
 If we multiply the terms together we get:
 

 Y = X3 -7X + 6

 
 The biggest exponent is a 3, 
 and the term with that exponent is positive.
 
 So that means the graph goes up on the right
 and down on the left, like Y = X 3.
 
 But the extra terms (-7X + 6), might give it a hitch in the middle.
 Kind of like this:

 

 
 Sometimes the graph line doesn't cross the X axis,
 but just touches it like this instead:
 

 

 What would the factors of this one look like?
 
 We need to get the graph to just touch the X axis where X = 1.
 
 There is a trick to this (big surprise, eh?)
 The factor that would make the graph
 cross the X axis at X - 1 is (X - 1).
 
 To make the graph just touch the X axis but not cross it,
 use that factor TWICE!
 
 That means the equation that goes with this graph:
 

 

 
 Is:

 Y = (X + 3)(X - 1)(X - 1)

 
 Sometimes we write this equation like this:
 

 Y = (X + 3)(X - 1)2

 
 Example:
 Draw the graph of ...

 Y = (X + 1)(X + 1)(X + 3)

 
 Y = 0 when X = -1 (the double factor) or when X = -3.
 If we multiply these factors together we get:
 

 Y = X3 + 5X2 + 7X + 3

 
 The biggest exponent is a 3. The term it is on is positive.
 Put all this together and we get ...

 

 
 Now just connect the lines and dots ...

 

 
 This double factor stuff may seem all new, but it's not.
 The very first equation we looked at in the second Algebra stuff
 was one of those "double factor" things.
 
 Oh yeah???
 
 Yeah!! It was:

 Y = X2

 

 If we want to be really technical,
 we can write it like this:
 

 Y = (X - 0)2    or    Y = (X - 0)(X - 0)

 
 It's a double factor equation that touches the X axis when X = 0.
 
 If you multiply it out, you get:

 Y = X2 (big surprise, eh?)

 
 The biggest exponent is a 2. That makes the graph a parabola.
 The term with the biggest exponent is positive,
 that means the parabola opens upward.
 
 Now we can draw the graph ...
 

 

 
 Hmmm, I wonder what would happen if we had a factor 
 show up THREE times?
 

   copyright 2005 Bruce Kirkpatrick

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