I need 2 quartic+ functions that have a complez zero.
*Complex
if a quartic has one complex zero, it will have two. They come in conjugate pairs. An easy way to make some up is to have two quadratic factors with negative discriminants:
(x^2+4x+10)(3x^2-8x+7)
= 3x^4+4x^3+5x^2-52x+70
you can easily make up others.
To find two quartic functions that have complex zeros, we first need to understand the properties of complex numbers and how they relate to polynomial functions.
A quartic function is a polynomial of degree 4, which means it can have a maximum of 4 complex zeros. Complex numbers consist of a real part and an imaginary part, represented as "a + bi," where "a" is the real part and "bi" is the imaginary part.
To create quartic functions with complex zeros, we can use the following general form:
f(x) = (x - z1)(x - z2)(x - z3)(x - z4)
Here, z1, z2, z3, and z4 represent the complex zeros of the function.
Let's consider two examples:
1. f(x) = (x - 2i)(x + 2i)(x - 3i)(x + 3i)
In this example, we have complex zeros: 2i, -2i, 3i, and -3i.
2. f(x) = (x - (1 + 2i))(x - (1 - 2i))(x - (2 + 3i))(x - (2 - 3i))
Here, the complex zeros are: (1 + 2i), (1 - 2i), (2 + 3i), and (2 - 3i).
Note that there are infinitely many possibilities for quartic functions with complex zeros, and these examples are just two illustrative instances.
To generate more examples, you can pick any complex numbers for z1, z2, z3, and z4 and plug them into the general form of a quartic function mentioned above.
Remember that complex zeros always come in conjugate pairs. If a complex number a + bi is a zero of a polynomial, its conjugate a - bi will also be a zero.
I hope this explanation helps you understand how to create quartic functions with complex zeros!