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