# 2x^3+x^2-18x-9 find maximum of zeros and list all possible zeros, use synthetic division to find all zeros, express (fx) as a product of linear factors

embizze | High School Teacher | (Level 1) Educator Emeritus

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Given the polynomial `2x^3+x^2-18x-9` :

(1) By the fundamental theorem of algebra there can be at most 3 real roots/zeros

(2) The possible rational roots are of the form `p/q` where p is a factor of the constant term and q a factor of the leading coefficient.

The possible rational roots are `+-1,+-3,+-9,+-1/2,+-3/2,+-9/2`

(3) Using synthetic division we guess at the first root. Luckily we guess correctly with 3 (so x-3 is a factor)

3 |  2   1   -18   -9
6    21    9
----------------
2   7   3     0

Repeating (the only possible rational roots now are `+-1,+-3,+-1/2,+-3/2`

Guessing -3 (so x+3 is a factor) we get:

-3|  2   7   3
-6 -3
-----------
2    1

Then (2x+1) is a factor so `x=-1/2` is a root.

(4) Thus we can express the polynomial as (x+3)(x-3)(2x+1)

** If you were not instructed to use synthetic division you might see:

`2x^3+x^2-18x-9`

`=x^2(2x+1)-9(2x+1)`

`=(2x+1)(x^2-9)`

`=(2x+1)(x+3)(x-3)` as above.