Refer to the polynomial function, h(x) = -3x^2 (5x + 5)(x - 1) of Finding the Zeros of a Polynomial Function. Describe and provide an explanation for the end behavior of h(x).

Answer 1

The polynomial function h(x) = -3x^2 (5x + 5)(x - 1) can be simplified further.

First, we can notice that -3x^2 can be factored out from the expression.

h(x) = -3x^2 (5x + 5)(x - 1)
= -15x^3 - 15x^2(x - 1)

Next, we can expand the expressions (5x + 5) and (x - 1) using the distributive property:

h(x) = -15x^3 - 15x^2(x - 1)
= -15x^3 - 15x^3 + 15x^2
= -30x^3 + 15x^2

So, after simplification, the polynomial function h(x) = -30x^3 + 15x^2.

Now, let's analyze the end behavior of this polynomial function.

As x approaches negative infinity (x → -∞), the term -30x^3 dominates the expression, since the coefficient -30 is multiplied by a large negative number. Thus, h(x) becomes more and more negative as x approaches negative infinity, resulting in the end behavior of h(x) being that the function approaches negative infinity as x approaches negative infinity.

Similarly, as x approaches positive infinity (x → +∞), the term -30x^3 again dominates the expression, but this time multiplied by a large positive number. Therefore, h(x) becomes more and more positive, approaching positive infinity as x approaches positive infinity. Hence, the end behavior of h(x) is that the function approaches positive infinity as x approaches positive infinity.