how to find the x intercepts for f(x) = 3x^3+7x^2+3x+1
please show your steps
I guess you didn't accept my previous reply to your question.
Well, there is one intercept at x = -1.899
rounded off to the third decimal.
As I told you before, it does not factor with rational numbers,
if you solve 3x^3+7x^2+3x+1 = 0 you get the real root that I stated, plus two imaginary roots.
The x intercepts are where y = f(x) = 0
I don't see any easy solutions here. Whatever roots (x-values) there are will be negative. I suggest a graphical or an iteration solution. One root is approximately x = -1.90
The exact solution is not difficult to obtain, though.
The equation is
x^3+7/3 x^2+x+1/3 = 0
Get rid of the quadratic term by substituting x = y - 7/9:
y^3 - 22/27 y + 362/729 = 0 (1)
This can be solved by comparing with the identity:
(a+b)^3 = a^3 + 3 a^2 b + 3 a b^2 + b^3
You can rewrite this as:
(a+b)^3 = 3ab(a+b) + a^3 + b^3
This means that a solution of the equation:
y^3 - 3ab y - (a^3 + b^3) = 0
is y = a + b
So, to solve (1) we can extract a and b:
3 a b = 22/27 (2)
a^3 + b^3 = -362/729 (3)
Take the third power of (2) and define
A = a^3, B = b^3:
A B = 22^3/3^12 (4)
And (3) can be written as:
A + B = -362/3^6 (5)
Solving (4) and (5) amounts to solving a quadratic equation, A and B are then the two solutions:
A = -181/3^6 + 1/81 sqrt(273)
B = -181/3^6 - 1/81 sqrt(273)
The real solution is then obtained as:
y = a + b
with
a = -[181/3^6 - 1/81 sqrt(273)]^1/3
b = -[181/3^6 + 1/81 sqrt(273)]^1/3
The complex solutions are obtained from the other cube roots of A and B. If we multiply a by exp(2 pi i n/3) to obtain another cube root of A, then we must multiply b by exp(-2 pi i n/3), because eq. (2) needs to be satisfied.
To find the x-intercepts of a function, we need to determine the values of x for which the function equals zero. In other words, we need to find the roots of the equation f(x) = 3x^3 + 7x^2 + 3x + 1.
Step 1: Set f(x) equal to zero:
3x^3 + 7x^2 + 3x + 1 = 0.
Step 2: Since the equation is a cubic polynomial, it may not be easily factored or have rational roots. Therefore, we can use numerical methods to estimate the roots. A common method is the Newton-Raphson method.
Step 3: Start with an initial guess for the root, denoted as x1. It is helpful to choose a value close to the expected root, based on observations of the function's graph or knowledge of the problem.
Step 4: Apply the Newton-Raphson method. The formula for finding the next approximation, denoted as x2, is:
x2 = x1 - f(x1) / f'(x1),
where f'(x) is the derivative of the function.
Step 5: Repeat step 4 until the desired level of accuracy is achieved or until the computed value stops changing significantly.
Step 6: The final obtained value(s) of x will be the x-intercept(s) of the function.
Applying these steps to the given function f(x) = 3x^3 + 7x^2 + 3x + 1 requires the use of numerical methods or a graphing calculator, as the function does not factor easily.