Homework Help

Find all the zeros of f(x). `f(x) = 3x^4 + 11x^3 + 11x^2 + x - 2` List answers from...

user profile pic

kristenmarieb... | Student, Grade 10 | (Level 1) Valedictorian

Posted July 24, 2013 at 9:46 PM via web

dislike 1 like

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.

1 Answer | Add Yours

user profile pic

mjripalda | High School Teacher | (Level 1) Senior Educator

Posted July 25, 2013 at 2:58 AM (Answer #1)

dislike 1 like

`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` .

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes