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Answers to the Sample Test!!

1) Use the remainder from each synthetic substitution to find the value( remainder theorem).
           Answers are:
                                     a) -39
                                     b) 86
                                     c) 19/4 or 4 and 3/4
                                     d) 15/4 or 3 and 3/4

2) Since the leading coefficient is 2 and the linear term is -5 and p(0) = 7 this means the polynomial is in the form: 2x³ + bx² - 5x + 7. Use the fact that p(2) = 21 to get the function.
21 = 16 + 4b - 10 + 7
21 = 4b + 13
8 = 4b
b = 2
The function is p(x) = 2x³ + 2x² - 5x + 7
Now use synthetic division to find p(-1)
Answer is 12

3) a) -4
     b) 9
     c) 0
     d) -7

4) Dividing by 1 gives you        1 -9 26 -24 0
     Dividing by 3 gives you        1 -6 8 0 This represents the function
                                                x² - 6x + 8 which factors to:
                                                (x - 2)(x - 4)
The other zeros are 2 and 4

5) a)

             A(x) = x(900- 3x)/ 2 = 450x - (3/2)x²
                    The vertex point happens at -b/2a = -450/ -3 = 150
                    The dimensions of the maximum area is 150 by 225 which yields an area of 33750 square feet.

7) Number of passengers = (2000 - 200x)
rate = (20 + 5x)
x represents the number of nickels to increase the fare.

R(x) = (2000 - 200x)(20 + 5x) = 40000 + 6000x - 1000x²
The maximum happens at -b/2a = -6000/-2000 = 3
Thus the rate should increase by 3 nickels or 15 cents, raising the rates to 35 cents

8) a)     x²(3x - 16) - 4(3x - 16) = 0
            (x² - 4)(3x - 16) = 0
            (x - 2)(x + 2)(3x - 16) = 0
The zeros are: 2, -2, 16/3

    b)    Let y = x²
            y² - 6y - 27 = 0
            (y - 9)(y + 3) = 0
            y = 9 or y = -3
            Substitute this into the top equation to find x
Thus x = 3, -3,

c) 2, 3, 1

d) 2, -1, -1 + 2i, -1 - 2i

9) a) If 2 + 3i is a factor, so is 2 - 3i. Using the imaginary roots, the sum = 4 and product = 13. Thus,
                                                          (x - 4)(x² - 4x + 13) =
                                                           x³ - 8x² + 29x - 52

    b) If 3 - 5i is a root, so is 3 + 5i. The sum is 6 and the product = 34. Thus, the equation is:
                                                          x² - 6x + 34

    c) The sum of the imaginary roots = 2 and the product = 2.
         The sum of the irrational roots = 4 and the product = 1
         Thus the equation is: (x² - 2x + 2)(x² - 4x + 1) Multiplying yields:
                                         x⁴ - 6x³ + 11x² - 10x + 2

10) a) sum = -4/-2 = 2 product = 7/2

b) sum = 3/4 product = -2/4 = -1/2

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