# 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.

lemjay | High School Teacher | (Level 3) Senior Educator

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`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` .