6. What are the zeroes for the

function

f (x) = 2x3 + 12x - 10x2?

x = 0, x = 5 + √13 and x = 5 - √13.

Step-by-step explanation:

f (x) = 2x^3 + 12x - 10x^2 can and should be rewritten in descending powers of x:

f (x) = 2x^3 - 10x^2 + 12x

This, in turn, can be factored into f (x) = x· (x² - 10x + 12).

Setting this last result = to 0 results in f (x) = x· (x² - 10x + 12).

Thus, x = 0 is one root. Two more roots come from x² - 10x + 12 = 0.

Let's "complete the square" to solve this equation.

Rewrite x² - 10x + 12 = 0 as x² - 10x + 12 = 0.

a) Identify the coefficient of the x term. It is - 10.

b) take half of this result: - 5

c) square this last result: (-5) ² = 25.

d) Add this 25 to both sides of x² - 10x + 12 = 0:

x² - 10x + 25 + 12 = 0 + 25

e) rewrite x² - 10x + 25 as the square of a binomial:

(x - 5) ² = 13

f) taking the sqrt of both sides: x - 5 = ±√13

g) write out the zeros: x = 5 + √13 and x = 5 - √13.

The three roots are x = 0, x = 5 + √13 and x = 5 - √13.