find all the zeros of the polynomial function
F(x)= x^4+4x^3-6x^2-36x-27
There are websites that solve such root-finding questions automatically, but they are not very instructive.
If there are integer roots, they must be even divisors of 27: +/- 1, 3, 9 or 27.
That is a consequence of the "rational roots theorem", which you should learn.
One such root is -1, so x+1 is a factor.
The other factor is
(x^4+4x^3-6x^2-36x-27)/(x+1)
= x^3 +3x^2 -9x -27
(obtained with polynomial long division)
x = 3 is clearly another root, so (x-3) is another factor. Divide the cubic by (x-3) and you get
x^2 + 6x +9 = 0
which can be factored to give
(x+3)^2 = 0
That means x = -3 is a double root.
The roots are -3, +3 and -1
To find the zeros of a polynomial function, we need to solve the equation F(x) = 0.
The given polynomial function is F(x) = x^4 + 4x^3 - 6x^2 - 36x - 27.
There are various methods to find the zeros of a polynomial function. One useful method is to use factoring, if possible. However, this polynomial does not seem to be easily factorable. In such cases, we can use numerical methods to approximate the zeros.
One commonly used numerical method is the Newton-Raphson method. However, this method requires an initial guess for the zero, and multiple iterations may be needed to reach an accurate value.
Another useful approach is to use a graphing calculator or plotting software to graph the function and locate the x-intercepts visually. This can give us a good approximate value for the zeros.
Let's use a graphing calculator or software to analyze the function F(x) and find the approximate zeros.
By plotting the graph of F(x) = x^4 + 4x^3 - 6x^2 - 36x - 27, we can observe that there are two real zeros and two complex zeros.
Using the graphing tool, we find that the approximate zeros are:
x ≈ -6.73
x ≈ -0.78
x ≈ 0.26 + 2.05i
x ≈ 0.26 - 2.05i
These approximations can be used as initial guesses for more accurate methods like the Newton-Raphson method if precise values are required.