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?
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.
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
thank you for both your help, i think i got it now!
To solve the inequality n^3 - 7n + 6 < 0 using the Rational Root Theorem, you need to follow these steps:
Step 1: Identify the coefficients of the polynomial. In this case, the coefficients are 1, -7, and 6.
Step 2: Find the possible rational roots. The Rational Root Theorem states that the rational roots (if they exist) must be of the form p/q, where p is a factor of the constant term 6, and q is a factor of the leading coefficient 1. The factors of 6 are ±1, ±2, ±3, and ±6, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1, ±2, ±3, and ±6.
Step 3: Test the possible roots using synthetic division or substitution. Substitute each possible root into the original inequality and check if it holds true.
Let's begin with the first possible root, n = 1:
Substituting n = 1 into the inequality:
1^3 - 7(1) + 6 < 0
1 - 7 + 6 < 0
0 < 0
Since the inequality does not hold true for n = 1, it is not a root. Repeat this process for the other possible roots: n = -1, n = 2, n = -2, n = 3, n = -3, n = 6, and n = -6.
By testing each possible root, you will find that n = -1 and n = 2 are the roots that satisfy the inequality.
Hence, the solution to the inequality n^3 - 7n + 6 < 0 is -1 < n < 2.