Please show me how to work out this problem. I know the roots are -1/3, -1/2, and 3, but I don't know how to work the problem by hand to find the answer. Please help so I know how to solve this kind of problem in the future!
That's correct. Now observe that all the coefficients
can be divided by -4, so we do that:
If there is any rational root, its numerator must divide
evenly into the absolute value of the constant term, and
its denominator must divide evenly into the absolute value
of the coefficient of the leading term (the term with the
largest exponent.
So the possible numerators are 1 and 3, for these are the
only integers which will divide evenly into 3.
the possible denominators are 1, 2, 3 and 6, for these are the
only integers which will divide evenly into 6.
So all possible rational (common fraction) roots are
±, ±, ±, ±, ±, ±, ±, ±.
Which reduce to:
±, ±, ±, ±, ±, ±, ±, ±.
Eliminating the duplicates, all possible rational roots are:
±, ±, ±, ±, ±, ±.
First we try the easiest possible root 1, using synthetic division:
1| 6 -13 -14 -3
| 6 -7 -21
6 -7 -21 -24
That doesn't leave a 0 remainder so 1 is not a root.
So we try the next easiest possible root -1, using synthetic division:
-1| 6 -13 -14 -3
| -6 19 -5
6 -19 5 -8
That doesn't leave a 0 remainder either so -1 is not a root.
So we try the next easiest possible root 3, using synthetic division:
3| 6 -13 -14 -3
| 18 15 3
6 5 1 0
That leaves a 0 remainder so 3 is a root. So we have now
factored the left side of:
as
Now we factor the second parentheses:
Now we use the zero factor property
to set each factor = 0
gives the solution
gives the solution
gives the solution
Edwin