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 2x3−7x2−46x−21=0?

1 Answer
Alan P.
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 ∈{−3,7,−12}

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:
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We notice that synthetic division of 2x3−7x2−46x−12
by (x+3) gives 2x2−13x−7 with a Remainder of 0

So 2x3−7x2−46x−12=(x+3)(2x2−13x−7)

We can then factor ((2x2−13x−7) as
XXX(2x+1)(x−7)

So 2x3−7x2−46x−12=(x+3)(2x+1)(x−7)
and
the roots are:
XXX{−3,−12,+7}
(the roots are the values of x that make the factors =0)

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