Find all rational roots for f(x)=x^3-7x^2-6

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

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Find all rational roots for `f(x)=x^3-7x^2-6` :

By a corollary of the fundamental theorem of algebra there are 3 roots in the complex numbers -- there are either exactly one real root or three real roots.

By the rational root theorem the only possible rational roots are `+-1,+-2,+-3,+-6`

We could try all eight possible roots. Using synthetic division we try -1:

-1  | 1   -7    0   -6
    ----------------
        1  -8    8   -14

Since the signs of the coefficients of the quotient alternate, there are no rational roots smaller than -1.

We are left to try 1,2,3,6:

f(1)= -12

f(2)=-26

f(3)=-42

f(6)=-42 So there are no rational roots.

Using a calculator, the only real root is `x~~7.1184092`

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** If the problem was `f(x)=x^3-7x-6` then there are three rational roots: -1,-2,3. **
    

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