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?

Feb 3, 2016

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

Explanation:

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

We notice that synthetic division of $\textcolor{g r e e n}{2 {x}^{3} - 7 {x}^{2} - 46 x - 12}$
by $\left(x + 3\right)$ gives $\textcolor{b l u e}{2 {x}^{2} - 13 x - 7}$ with a Remainder of $\textcolor{red}{0}$

So $\textcolor{g r e e n}{2 {x}^{3} - 7 {x}^{2} - 46 x - 12} = \left(x + 3\right) \left(\textcolor{b l u e}{2 {x}^{2} - 13 x - 7}\right)$

We can then factor ((color(blue)(2x^2-13x-7)) as
$\textcolor{w h i t e}{\text{XXX}} \left(2 x + 1\right) \left(x - 7\right)$

So $\textcolor{g r e e n}{2 {x}^{3} - 7 {x}^{2} - 46 x - 12} = \left(x + 3\right) \left(2 x + 1\right) \left(x - 7\right)$
and
the roots are:
$\textcolor{w h i t e}{\text{XXX}} \left\{- 3 , - \frac{1}{2} , + 7\right\}$
(the roots are the values of $x$ that make the factors $= 0$)