# Using complete sentences, explain how to find the zeros of the function f(x) = 2x^3-9x+3. Explain how to find the zeros of the function f(x)= (x-6)^3Algebra 2 Honors homework

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### 1 Answer

You need to create the set of divisors of constant term such that:

`D_3 = {+-1 ; +-3}`

You need to create the set of divisors of leading coefficient such that:

`D_2 = {+-1 ; +-2}`

You may find the zeroes of the function in the set that consists from all the terms `D_3/D_2` such that:

`D_3/D_2 = {+-1/2 +- 3/2}`

You need to consider `x = 1/2` such that:

`f(1/2) = 2*(1/2)^3 - 9/2 + 3 = 1/4 - 9/2 + 3`

`f(1/2) = (1-18+12)/4 != 0`

You need to consider `x = -1/2` such that:

`f(-1/2) =(-1-18+12)/4 != 0`

You need to consider `x = 3/2` such that:

`f(3/2) = 2*(3/2)^3 - 27/2 + 3 =27/4 - 27/2 + 3`

`f(3/2) = (27 - 54 + 12)/4 != 0`

Notice that none of these fractions satisfy the equation `f(x) = 0` .

You need to draw the graph and you need to check where the garph intercepts x axis such that:

Notice that the equation `f(x) = 0` has three real roots since the red curve intercepts x axis three times, hence, the first root is `x_1 in (-2.5,-2)` , the second root is `x_2 in (0,0.5)` and the thirs root is `x_3` ```~~` 1.9.

You need to find the zeroes of the function `f(x) = (x-6)^3,` hence, you need to solve the equation f(x) = 0 such that:

`(x-6)^3 = 0 => (x-6)(x-6)(x-6) = 0`

`x - 6 = 0 => x = 6`

**Notice that the equation `(x-6)^3 = 0` has the root x=6 three times, hence, you may say that the polynomial `(x-6)^3 = 0` has the root x=6 of multiplicity 3.**