Using the rational zeros theorem to find all zeros of a polynomial

Algebra
Tutor: None Selected Time limit: 1 Day

The function below has at least one rational zero.  Use this fact to find all zeros of the function
      f(x)=3x^3-7x^2-19x+7
if more than one zero, separate with commas.  Write exact values, not decimal approximations

Oct 18th, 2015

Thank you for the opportunity to help you with your question!

The rational roots theorem tells us that any rational root must be of the form p/q, where p is a factor of 7 and q is a factor of 3. This gives us the possibilities +-7, +-1/3 or +_1. Testing these we get that the rational root is 1/3. Now, the factor theorem tells us that we can divide the original polynomial by qx-p if p/q is a factor.

Dividing 3x^2-7x^2-19x+7 by 3x-1 gives us x^2-2x-7.

To find the remaining two roots all we have to do is complete the square.

Since x^2-2x-7=0,      add 8 to both sides

so x^2-2x+1=8

(x-1)^2=8

so x=+- rad(8)+1

x=r 1+_rad(8)

x= 1+_2rad(2)

with the original rational root still 1/3


Please let me know if you need any clarification. I'm always happy to answer your questions.
Oct 18th, 2015

what is meant by the underscore

Oct 18th, 2015

im confused by what the final answer is

Oct 18th, 2015

+_ means +- i.e plus or minus.

the final answer is 1/3, 1+- rad(2)  

rad means square root of.

Oct 18th, 2015

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