Question

How do you use the rational root theorem to find the roots of `x^3-9x^2+14x=48`?

Answer 1

Test all possible values of `x in {+- ("factors of " 48)/("factors of "1)}`
Any that satisfy the equation are roots of the equation.
`color(white)("XXX")`(The only Real root is at `x=8` in this case).

Explanation:

The candidates for rational roots of `color(red)(1)x^3-9x^2+14x-color(blue)(48)=0`
(notice, I've slightly re-arranged the equation)
are `+-("factors of "color(blue)(48))/("factors of "color(red)(1))`
`color(white)("XXXXXXXXXXXX")= +- (1,2,3,4,6,8,12,16,24,48)`

Therefore there are `20` candidates to test!
(It is useful if you can set this up as a spreadsheet or computer program.)

Here are the evaluations (using synthetic substitution/division) for the first `8` candidates:

This gives us `x=8` as a zero of the given equation.

Not of the subsequent factors gave a result of `0` (Take my word for it).

It might be noted that treating these entries as synthetic division gives:
`color(white)("XXX")x^3-9x^2+14x-48 = (x-8)(x^2-x+6)`
and taking the discriminant of `(x^2-x+6)` shows us that there are no further Real roots (i.e. no other Real zeros)