Question

List all possible zeros

Answer 1

values for x up to 7 decimals where f(x) becomes zero:

0.1321827

-1.3160914+3.1009404i

-1.3160914-3.1009404i

and

`i = √( - 1)`

Answer 2

The possible zeroes are -3 , -5/3 and 1/2

Explanation:

* To find the zeroes you must let f(x) = 0

* First step look at the numerical term -15 then find two numbers multiply by each other and = -15⇒[ (5×-3) (-5×3) (-1×15) (1×-15)] and let x = one of them

* Second step Chose on of them and substitute x by it, I will chose -3

* f(-3) =
`6(-3^(3))+25(-3^(2))+16(-3)-15=-162+225-48-15=0`

∵ f(x) = 0 when x = -3

∴ x + 3 is a factor of f(x)

* Third step divide f(x) by its factor to get quadratic and factorize it

`(6x^(3)+25x^(2)+16x-15  )/(x+3)=6x^(2)(7x^(2)+16x-15 )/(x+3)`

`(7x^(2)+16x-15 )/(x+3)=7x(-5x-15)/(x+3)`

`(-5x-15)/(x+3)=-5`

∴ f(x) = (x + 3)(
`6x^(2)+7x-5`⇒factorize the quadratic

∵f(x) = (x + 3)(2x - 1)(3x + 5)

∵ f(x) = 0

∴ x + 3 = 0 ⇒ x = -3

∴ 2x - 1 = 0 ⇒ x = 1/2

∴ 3x + 5 = 0 ⇒ x = -5/3

∴