Algebra 2 Rational Zeros Theorem
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Cross The Line
Rational Zeros Theorem

 

 It is not always so easy to find the places

 where the graph line crosses the X axis.
 Those are the places where Y = 0.
 It would be great if there was a way to figure out 
 where these places were.
 
 WE CAN!
 Sometimes ...
 
 Say we have some nasty equation like:
 

 Y = 2X3 - 3X2 - 2X + 3

 
 Does this equation cross the X axis?
 WHO KNOWS???
 
 We can guess at X values that MIGHT make Y = 0.
 We can graph the equation and hope we can tell
 where the line crosses the X axis.
 OR ...
 We can use a new trick.
 
 For this trick to work, all of the coefficients (the numbers)
 must be integers.
 Integers are stuff like 6 and -2, and 0.
 Things that are not integers are things like 1/2 and 6.2 and p.
 
 So since all of the numbers in this equation ARE integers,
 we can use the new trick.
 Here goes ...
 
 First we build a fraction.
 Take the term that has no X and divide it into prime factors.
 In this problem, the term with no X is a +3.
 The prime factors of +3 are:

  3 = 1 3 so we have a 1 and a 3

 
 These will be the numerators (numbers on top of a fraction),
 for this trick.
 
 Now take the coefficient of the term with the biggest exponent.
 In this problem, the term with the biggest exponent is 2X 3
 so we want the 2.
 Divide it into prime factors ...
 

  2 = 1 2 so we have a 1 and a 2

 
 These will be the denominators (numbers on the bottom of a fraction)
 for our trick.
 
 Our numerators are 1 and 3
 Our denominators are 1 and 2
 List all of the fractions that have 1 or 3 on top
 and 1 or 2 on the bottom.
 

 

 
 If the equation crosses the X axis at any rational number,
 it will happen at one or more of these places:
 

 

 
 Great, now what?
 Now we use synthetic division.
 When the equation crosses the X axis, Y = 0.
 That means the remainder we get with synthetic division
 will be ZERO!
 
 Most people don't use synthetic division very much
 so take a minute to look up how to do it if you want.
 
 Back already? OK let's do it.
 
 Check the -3:

 

 
 Check some more numbers:

 

 
 So the equation line crosses the X axis at X = -1
 Let's see if it crosses any where else that we can find ...
 

 

 
 S0 the equation line also crosses the X axis at X = 1.
 Now we get to so the hard ones ...
 

 

 

 

 
 So the equation line crosses the X axis at
 X = -1, X = 1, and X = 3/2
 
 Remember that to use this trick
 all of the numbers in the equations MUST be integers,
 but the numbers we get for X don't have to be integers.
 
 There is another cute trick that we can use.
 Since X = -1, X = 1, and X = 3/2 are all values that make Y = 0.
 (X - (-1)), (X - 1), and (X - 3/2) are all factors of Y = 2X 3 - 3X 2 -2X + 3
 
 Are they ALL the factors of the equation?
 Let's find out.
 Multiply them all together and see what you get ...

 

 
 That's close but something is missing.
 We want:

 2X3 - 3X2 - 2X + 3

 and we have:

 

 
 See what's missing?
 We need to multiply each term by 2.
 So that's the missing factor, a 2!

 

 That means:

 Y = 2X3 - 3X2 - 2X + 3

 Factors as:

 

 
 And the ONLY RATIONAL NUMBER places that the equation line
 crosses the X axis are:

 

 
 You keep saying "RATIONAL NUMBERS," why?
 
 Because the graph line could cross the X axis 
 at an irrational number like p
 If it does, this trick won't help us find it.
 

   copyright 2005 Bruce Kirkpatrick

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