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State all of the possible rational zeros for the function f(x) = 2x^3 + 2x^2 - x + 16 using the Rational Zero Test Simplify fractions if necessary. 

Given the function f(x) = -x(x+1)^3(x-5)(x-3)^4 , name the zeros and wether or not there would be a bounce or a pass at each of the zeros on the graph.

Re-write the following function f(x) = -3x^2 + 12x - 2 in vertex form. Remember, this is the form f(x) = a(x - h)^2 + k.

 Apr 19, 2020
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1)  The Rational Zero Test says that any possible rational zero is a fraction whose numerator is a divisor of the constant term

      (in this case, a divisor of 16) and the denominator is a divisor of the coefficient of the first term (in this case, 2).

      Also, all the negative must also be considered.

      The divisors of 16 are 16, 8, 4, 2, 1.  

      The divisors of 2 are 2, 1.    

      So, for this problem the possibilities are:  16/1, -16/1, 8/1, -8/1, 4/1, -4/1, 2/1, -2/1, 1/1, -1/1

                                                               and:  16/2, -16/2, 8/2, -8/2, 4/2, -4/2, 2/2, -2/2, 1/2, -1/2

      Of course, some of these are repeats.

 

      Try these one at a time until you find one that makes the function zero. If it makes the function zero, it will be a root; if

       it doesn't make the function zero, it isn't a root.

 

       For example:  f(x)  =  2x3 + 2x2 - x + 16     --->     f(16)  =  2(16)3 + 2(16)2 - (16) + 16  =  8704

       This didn't make the function zero, so it isn't a root.

       So, you get to try the next number.

       Keep trying until you get one that works or until they all fail.

       Once you get one that works, divide the function by that root (for instance, if 5 works, divide by (x - 5)).

       If you can factor the result, do so; that's easier than using the RZT.    

       If you can't factor the result, use the RZT on this new function.

 

2)  Every factor that has an even exponent bounces; every factor that has an odd exponent passes through.

 

3)  f(x)  =  -3x2 + 12x - 2

     Bring the constant term to the other side:            f(x) + 2  =  -3x2 + 12x

     Factor out the coefficient of the squared term:    f(x) + 2  =  -3(x2 - 4x)

     Complete the square by dividing the coefficient of the x-term by 2 and squaring that answer:

         -4 / 2  =  -2   --->   (-2)2  =  4

     Add that inside the parentheses -- note that you really are adding a value of -3 x 4  =  -12, so you'll have

     to add that to the other side as well:            f(x) + 2 - 12  =  -3(x2 - 4x + 4)

                                                                                 f(x) -10  =  -3(x - 2)2

                                                                                        f(x)  =  -3(x - 2)+ 10

 Apr 19, 2020

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