Create polynomial functions using the given information:

A degree 7 polynomial with at most 6 positive zeros and at most 1 negative zero.

a) $ f(x) = x^7 - 5x^6 + 10x^5 - 10x^4 + 5x^3 - x^2 + x - 1 $

b) $ f(x) = x^7 - 4x^6 + 6x^5 - 5x^4 + 4x^3 - 3x^2 + 2x - 1 $

c) $ f(x) = x^7 - 6x^6 + 15x^5 - 20x^4 + 15x^3 - 6x^2 + x - 1 $

d) $ f(x) = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1 $

Polynomial functions of degree 7 with at most 6 positive zeros and 1 negative zero can be designed by adjusting coefficients and constants in a way that satisfies these conditions. Example functions include f(x) = x^7 - 5x^6 + 10x^5 - 10x^4 + 5x^3 - x^2 + x - 1, f(x) = x^7 - 4x^6 + 6x^5 - 5x^4 + 4x^3 - 3x^2 + 2x - 1, and others.

Explanation:

To create polynomial functions of degree 7 with at most 6 positive zeros and at most 1 negative zero, you would need to structure your polynomial function in a specific way to satisfy these conditions. Here are some examples:

f(x) = x^7 - 5x^6 + 10x^5 - 10x^4 + 5x^3 - x^2 + x - 1
f(x) = x^7 - 4x^6 + 6x^5 - 5x^4 + 4x^3 - 3x^2 + 2x - 1
f(x) = x^7 - 6x^6 + 15x^5 - 20x^4 + 15x^3 - 6x^2 + x - 1
f(x) = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1

Note that the highest power (degree) is 7, and the coefficients and constants are adjusted such that these polynomial functions can have at most 6 positive zeros and 1 negative zero.

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