# Find all the zeros of f(x). `f(x) = 3x^4 + 11x^3 + 11x^2 + x - 2` List answers from smallest to largest. If there is a double root, list it twice. Keep fractions in fractional form.

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### 1 Answer

`f(x)=3x^4 + 11x^3 + 11x^2 + x - 2`

To find the zeros of this, apply the Rational Zeros Theorem.

So, the possible zeros are `+-1, +-2, +-1/3, and +-2/3` .

To determine which of these are the zeros of the function, divide the polynomial by each zeros.

To do so, use synthetic division.

`-1` `|` `3` `11` `11` `1` `-2`

`-3` `-8` `-3` `2`

_____________________

`3` `8` `3` `-2` `0`

Since the last number is zero, then x=-1 is a zeros of f(x).

Then, divide the quotient by other possible zeros.

`-2` `|` `3` `8` `3` `-2`

`-6` `-4` `2`

__________________

`3` `2` `-1` `0`

Since the resulting last number is zero, then x=-2 is a zeros of f(x) too.

Then, divide the quotient again by the other possible zeros.

`1/3` `|` `3` `2` `-1`

`1` `1`

____________

`3` `3` `0`

So, x=1/3 is a zeros of f(x).

Then, divide by the other possible zeros.

`-1` `|` `3` `3`

`-3`

________

`3` `0`

Again, x=-1 is zeros of the function.

**Hence, the zeros of the function are:**

**`x= -1` , with multiplicity of 2**

**`x=-2` and**

**`x=1/3` .**