f(x)=6x^3+19x^2+8x-5
find all the zeros of the given function
If a cubic always try dividing by (x-1) and by (x+1) first to see if 1 or -1 is a root.
In this case:
x = -1 , 1/3 , -2 1/2
if all else fails:
http://www.solvemymath.com/online_math_calculator/algebra_combinatorics/polynomial_calculator/polynomial_roots.php
To find the zeros of a function, we need to solve the equation f(x) = 0. In this case, the function is f(x) = 6x^3 + 19x^2 + 8x - 5.
There are different methods to find the zeros of a polynomial function, such as factoring, using the rational root theorem, or using numerical methods like graphing or using a calculator.
Let's start by attempting to factor the polynomial. However, factoring a cubic polynomial is not always straightforward, especially when it does not contain any obvious common factors or known quadratic factors.
In such cases, we can proceed to try the rational root theorem as it can help us find potential rational roots (zeros) of a polynomial. The rational root theorem states that if a rational number p/q is a zero of a polynomial, then p must be a factor of the constant term, and q must be a factor of the leading coefficient.
In our case, the constant term is -5, and the leading coefficient is 6. Therefore, we need to consider all the factors of -5 and 6.
The factors of -5 are ±1 and ±5. The factors of 6 are ±1, ±2, ±3, and ±6.
Now, we can test each possible rational root by substituting them into the function f(x) = 6x^3 + 19x^2 + 8x - 5 and checking if the result is zero.
For example, let's start with the factor pairs (p, q) of (-5, 6):
- (p, q) = (1, 1)
- (p, q) = (1, -1)
- (p, q) = (-1, 1)
- (p, q) = (-1, -1)
- (p, q) = (5, 1)
- (p, q) = (5, -1)
- (p, q) = (-5, 1)
- (p, q) = (-5, -1)
Substituting these rational roots into the function f(x) = 6x^3 + 19x^2 + 8x - 5, we can check if any of them make the equation equal to zero.
For example, if we substitute x = 1, we have:
f(1) = 6(1)^3 + 19(1)^2 + 8(1) - 5
= 6 + 19 + 8 - 5
= 28
Since f(1) ≠ 0, x = 1 is not a zero of the polynomial.
By systematically checking all the potential rational roots, we can determine if any of them are zeros of the function. If a rational root is found, we can use polynomial division or synthetic division to factorize the polynomial further and find the remaining roots.
If none of the potential rational roots satisfy f(x) = 0, we can resort to using numerical methods like graphing the function or utilizing numerical algorithms to estimate the zeros.
Please note that finding the zeros of a cubic polynomial can be a complex process, and it is not always possible to find exact algebraic solutions.