The rational root theorem is explained here:
http://en.wikipedia.org/wiki/Rational_root_theorem
It tells you than one real root of the equation n^3-7n+6 = 0 must be of the form
+/- (1,2 or 3)/1 = +/1 3, 2 or 1. Obviously, +2 is one root.
Divide n-2 into your cubic and you get n^2 +2n -3 = 0
which factors to
(n+3)(n-1) = 0
which tells you that x=1 and x=-3 are also roots.
Now return to the inequality. Since n^3-7n+6> 0 for large n, it will be negative for x less than the largest root (2) but larger then the second largest root (1), because the sign of the function changes at each zero. There will be one other region where the function in negative. I will leave that for you to discover.
solve the inequality:
n^3-7n+6<0
it said to use the rational route theorem, but i don't quite understand how to use it. any help?
3 answers
I do not know this theorem BUT
Oh, Ho x=1 gives zero
so factor the thing
(x-1)(something) = x^3 -7x +6
do long division or synthetic division
(x-1)(x^2 + x - 6) <0
(x-1)(x+3)(x-2) = 0
so
It crosses the x axis at -3,1,2
for large - x, it is -
for large + x it is +
sketch that,
so it is negative when x < -3
and it is negative between 1 and 2 1<x<2
Oh, Ho x=1 gives zero
so factor the thing
(x-1)(something) = x^3 -7x +6
do long division or synthetic division
(x-1)(x^2 + x - 6) <0
(x-1)(x+3)(x-2) = 0
so
It crosses the x axis at -3,1,2
for large - x, it is -
for large + x it is +
sketch that,
so it is negative when x < -3
and it is negative between 1 and 2 1<x<2
thank you for both your help, i think i got it now!
1 answer