Find zeros:
f(x)=12x^(3)-107x^(2)-15x+24
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut38_zero1.htm#step2
Given:
f(x)=12x^(3)-107x^(2)-15x+24
Also if the leading coefficient (12) is positive, the function evaluates to -∞ at x=-∞ and to ∞ at x=∞. Thus it crosses the x axis at at least one place, or there is at least one real root.
In fact, application of Descarte's rule indicate that there are two changes of sign, so there are at least two positive roots. This also means that all three roots are real, since complex roots must come in pairs (conjugates).
Equate
f'(x)=36x²-214x-15
to zero, and solve for the maximum and minimum, if any.
f'(x)=0 at x=-0.07 and x=6.01
f(-0.07)=24.52 (maximum)
f(6.01)=-1326,
so there is a root close to 1 and probably another around -1.
By numerical methods, such as Newton's method, it should be possible to evaluate the zeroes as:
x=-0.531, x=0.417, and x=9.031
Check:
(x1*x2*x3)=2=24/12
(x1+x2+x3)=8.917 = 107/12
Thanks
To find the zeros of the function f(x) = 12x^3 - 107x^2 - 15x + 24, we need to find the values of x for which f(x) equals zero.
There are several methods to find the zeros of a polynomial function, but one commonly used method is the factor theorem combined with synthetic division.
Step 1: Factor out any common factors, if possible.
In this case, there are no common factors that can be factored out, so we move on to the next step.
Step 2: Use the factor theorem to find possible rational zeros.
The factor theorem states that if a polynomial function has a rational zero, it must be of the form p/q, where p is a factor of the constant term (in this case, 24) and q is a factor of the leading coefficient (in this case, 12).
The factors of 24 are ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24, and the factors of 12 are ±1, ±2, ±3, ±4, ±6, and ±12.
So, the possible rational zeros for f(x) are:
±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±8/1, ±12/1, and ±24/1, which simplifies to:
±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.
Step 3: Use synthetic division to test the possible zeros.
Using synthetic division, we divide f(x) by each possible zero to check if it results in a remainder of zero.
Starting with x = -1, we perform synthetic division as follows:
-1 | 12 -107 -15 24
|
|_________ -12 119 -104
Since the remainder is not zero, -1 is not a zero of f(x).
Similarly, we can perform synthetic division for each possible zero.
After checking all the possible zeros, we find that the zeros of f(x) = 12x^3 - 107x^2 - 15x + 24 are x = -0.5, x = 3/4, and x = 2.