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

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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

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