Go over this with me please, I forgot how to do it.
Solve the polynomial equation.
15x^3-119x^2-10x+16=0
nvm... i figured out how, you factor then solve :)
let f(x) = 5x^3 - 119x^2 - 10x + 16
now try some f(x) , where x = ±1, ±2, ±4, ±8 ...
hopefully to get a zero
took some some, but f(8) = 0
so (x-8) will be a factor
now by synthetic division , I got
15x^3-119x^2-10x+16
= (x-8)((15x^2 + x - 2) = 0
which factors further to
(x-8)(3x - 1)(5x + 2) = 0
x = 8, or x = 1/3 or x = -2/5
Sure! To solve a polynomial equation, like the one you've provided, we need to find the values of x that make the equation equal to zero. In this case, we have the equation:
15x^3 - 119x^2 - 10x + 16 = 0
To solve it, we can use various methods such as factoring, synthetic division, or the rational root theorem combined with the long division method.
Since this equation does not appear to have any easily identifiable factors, we will use the rational root theorem to check for possible rational roots first. According to the rational root theorem, any rational root of the equation must be of the form p/q, where p is a factor of the constant term (16) and q is a factor of the leading coefficient (15).
The factors of 16 are ±1, ±2, ±4, ±8, and ±16.
The factors of 15 are ±1, ±3, ±5, and ±15.
Applying the rational root theorem, we need to test all the possible combinations of p/q to see if they yield a root. By trying out all the possible combinations, we can use the remainder theorem to see if the polynomial is divisible by a given factor. If the remainder is zero, it means that the value of x is a root.
After trying the various combinations, in this case, there are no rational roots that satisfy the equation. This means we have to use numerical methods, such as factoring or graphing, to find the approximate solutions.
One common numerical method for finding approximate solutions is to graph the equation and look for the x-intercepts. Using graphing software or a graphing calculator, we can plot the equation and see where it crosses the x-axis. The x-values where the graph crosses the x-axis are the solutions to the equation.
Alternatively, if factoring is possible, it can also be used to find the solutions. However, factoring a cubic equation can be quite challenging in most cases.
To summarize, in this particular example, we were unable to find any rational roots using the rational root theorem. Therefore, we need to rely on numerical methods like graphing or factoring to find approximate solutions.