Question

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

Answer 1

The answer to your question is below

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