1. Consider the function f(x) = 5x^3 - 31x^2 - 129x + 27
a) verify that f(9) = 0. Since f(9) = 0, what is the factor?
b) find the remaining two factors
c) state all three zeros of the function

2. Use the factor theorem to determine if 3x - 4 is a factor of f(x) = 3x^2 + 2x - 8

The answer to your question is below

Step-by-step explanation:

Data

1.- f(x) = 5x³ - 31x² - 129x + 27

a) f(9) = 5(9)³ - 31(9)² - 129(9) + 27

   f(9) = 5(729) - 31(81) - 1161 + 27

   f(9) = 3645 - 2511 - 1161 + 27

   f(9) = 0

b.- I will use synthetic division

                        5   - 31  -129  + 27       9

                               45   126  -27                            

                        5     14     -3       0

    Trinomial = 5x² + 14x - 3

Factor               5x² + 15x - 1x - 3

                        5x(x + 3) - 1(x + 3)

b)                           (x + 3)(5x - 1)

                       x₂ + 3 = 0          5x₃ - 1 = 0

                       x₂ = -3                 x₃ = 1/5

c) The roots are x₁ = 9, x₂ = -3 and x₃ = 1/5

2.-  f(x) = 3x² + 2x - 8      factor = 3x - 4

                                     x    + 2

                       3x - 4    3x² + 2x - 8

                                   -3x² + 4x

                                     0    +6x - 8

                                           -6x  + 8

                                                      0  Remainder

As the remainder was "0", 3x - 4 is a factor of 3x² + 2x - 8

lockers 1, 4, 9, 16, 25, 40, and 49 are open

i ran through it by the other answer is wrong

answer: approximately 206 bones

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