Which of the following represents the zeros of f(x) = 6x³ - 35x² + 26x - 5?
-5,1/3,1/2
5,-1/3,1/2
5,1/3,-1/2
5,1/2,1/3

Solution:

Given, f(x) = 6x³ - 35x² + 26x - 5

From the options above, we know “5” is one of the factors of the function f(x) = 6x³ - 35x² + 26x - 5.

When x = 5

f(5) = 6(5)³ - 35(5)² + 26(5) - 5

f(5) = 750 - 875 + 130 - 5

f(5) = 0

Thus, (x - 5) is a factor of f(x) = 6x³ - 35x² + 26x - 5

When x = 1/2

f(1/2) = 6(1/2)³ - 35(1/2)² + 26(1/2) - 5

f(1/2) = 3/4 - 35/4 + 13 - 5

f(1/2) = 0.75 - 8.75 + 13 - 5

f(1/2) = 0

Thus, (x - 1/2) is a factor of f(x) = 6x³ - 35x² + 26x - 5

When x = 1/3 

f(1/3) = 6(1/3)³ - 35(1/3)² + 26(1/3) - 5

f(1/3) = 2/9 - 35/9 + 26/3 - 5

f(1/3) = 0

Thus, (x - 1/3) is a factor of f(x) = 6x³ - 35x² + 26x - 5

When x = -1/2

f(1/2) = 6(-1/2)³ - 35(-1/2)² + 26(-1/2) - 5

f(1/2) = -3/4 - 35/4 - 13 - 5

f(1/2) = -0.75 - 8.75 - 13 - 5 = -27.5

f(1/2) ≠ 0

Thus, (x + 1/2) is not a factor of f(x) = 6x³ - 35x² + 26x - 5

When x = -1/3 

f(1/3) = 6(-1/3)³ - 35(-1/3)² + 26(-1/3) - 5

f(1/3) = -2/9 - 35/9 - 26/3 - 5 = -17.79

f(1/3) ≠ 0

Thus, (x + 1/3) is a factor of f(x) = 6x³ - 35x² + 26x - 5

Therefore, the factors of f(x) = 6x³ - 35x² + 26x - 5 are 5, 1/2 and 1/3.

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Which of the following represents the zeros of f(x) = 6x³ - 35x² + 26x - 5?

Summary:

The zeros of f(x) = 6x³ - 35x² + 26x - 5 are 5, 1/2 and 1/3. Hence last option  is the correct answer.