Question

How do you list all possible rational roots for each equation, use synthetic division to find the actual rational root, then find the remaining 2 roots for `2x^3-7x^2-46x-21=0`?

Answer 1

Assume that one of the rational roots is integer;
then perform synth.div for each integer root of `21`
Final answer: roots `in {-3,7,-1/2}`

Explanation:

The integer factors of `21` are `{1,3,7,-1,-3,-7}`
Performing the synthetic division with `(x+f)`
for each `f` which is an integer factor of `21`:

We notice that synthetic division of `color(green)(2x^3-7x^2-46x-12)`
by `(x+3)` gives `color(blue)(2x^2-13x-7)` with a Remainder of `color(red)(0)`

So `color(green)(2x^3-7x^2-46x-12)=(x+3)(color(blue)(2x^2-13x-7))`

We can then factor `((color(blue)(2x^2-13x-7))` as
`color(white)("XXX")(2x+1)(x-7)`

So `color(green)(2x^3-7x^2-46x-12)=(x+3)(2x+1)(x-7)`
and
the roots are:
`color(white)("XXX"){-3,-1/2,+7}`
(the roots are the values of `x` that make the factors `=0`)