find all zeros of the function
6x^3+43x^2+6x+7
To find the zeros of a function, we need to set the function equal to zero and solve for x.
The given function is: 6x^3 + 43x^2 + 6x + 7
To solve for the zeros, we can use various methods, such as factoring, graphing, or using numerical methods like the Rational Root Theorem or Newton's method.
In this case, since the function is a cubic polynomial, it may not be easily factored. Therefore, let's use a numerical method.
One way to find the zeros is by using online calculators or software that can solve equations. However, if you want to understand the general process, we can use an iterative method like Newton's method.
Newton's method involves making an initial guess for a zero, and then using the formula:
x1 = x0 - (f(x0) / f'(x0))
Where x1 is the new approximation, x0 is the initial guess, f(x0) is the value of the function at x0, and f'(x0) is the derivative of the function at x0.
Let's start with an initial guess of x0 = 0.
To calculate x1, we need to evaluate the function and its derivative at x0:
f(x0) = 6x^3 + 43x^2 + 6x + 7
f'(x0) = d/dx (6x^3 + 43x^2 + 6x + 7)
Calculating the derivatives:
f'(x0) = 18x^2 + 86x + 6
Now, substitute the values into the Newton's method formula:
x1 = x0 - (f(x0) / f'(x0))
x1 = 0 - ((6x^3 + 43x^2 + 6x + 7) / (18x^2 + 86x + 6))
Calculate x1 using the above formula and substitute it back into the formula to calculate x2, and so on until you reach a desired level of accuracy.
Alternatively, you can use graphing tools or computer software to find the zeros of the function more quickly and accurately.
clearly you are intended to use the Rational Zeroes Theorem, and start guessing. A little synthetic division reveals that there are no rational roots at all.
So, the only possible roots will involve cube roots, and be very hard to find.
I suggest graphing it and approximating the value(s)
Since all the coefficients are positive, Descartes' Rule of Signs says that there are no positive real roots.
So, look for a negative root. Wolframalpha.com says it's at about -7.05