Answer :
Final answer:
The highest degree in the polynomial (the leading term) determines the behavior of the function as x approaches negative or positive infinity. If the leading term's power is even and its coefficient negative, or the power is odd and the coefficient is positive, the function will tend toward negative infinity as x does same. In the given options, the functions that satisfy these conditions are f(x) = -5x⁸ and f(x) = -2x⁵ - 3x + 26.
Explanation:
In the given question, we are examining the behavior of various polynomials as x approaches negative infinity. Generally, when dealing with polynomial functions, the term of highest degree (also called the leading term) dominates the behavior of the function as x approaches positive or negative infinity. The exponent of the leading term (its degree) and its leading coefficient will determine how the function behaves.
Thanks to the properties of even and odd functions, we can conclude that if the highest degree in the polynomial function is even and the leading coefficient is positive, as x approaches negative infinity, the function will tend to positive infinity. But if the highest degree is even with a negative leading coefficient, our function will tend to negative infinity. On the other hand, if the highest degree is odd and the coefficient is positive, the function will tend toward negative infinity, and vice versa if the coefficient is negative.
So, in this case, the functions where f(x) would approach negative infinity as x approaches negative infinity are: f(x) = -5x⁸ because the power is even and leading coefficient is negative, and also f(x) = -2x⁵ - 3x + 26 where the power is odd and leading coefficient is negative.
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