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

Answer 1

To analyze the end behavior of the polynomial function $h(x)=-3x^2(5x+5)(x-1)$, we start by identifying its leading term.

Step 1: Expand the function

1. First, we can simplify the factor $5x+5$ as $5(x+1)$.

2. Thus, we can rewrite the function: $$h(x)=-3x^2\cdot 5(x+1)(x-1)$$

3. Next, we can expand $(x+1)(x-1)$: $$(x+1)(x-1)=x^2-1$$

4. Combine everything: $$h(x)=-3x^2\cdot 5(x^2-1)=-15x^2(x^2-1)=-15(x^4-x^2)=-15x^4+15x^2$$

Step 2: Analyze the leading term

The leading term of the polynomial $h(x)=-15x^4+15x^2$ is $-15x^4$. The degree of the polynomial is 4, which is even, and the coefficient of the leading term is negative.

Step 3: Determine end behavior

The end behavior of a polynomial function is determined by its leading term as $x$ approaches positive or negative infinity:

1. As $x\to +\mathrm{\infty }$:

   • The leading term $-15x^4$ will dominate, and since it has a negative coefficient, $h(x)$ will approach $-\mathrm{\infty }$.

2. As $x\to -\mathrm{\infty }$:

   • Again, the leading term $-15x^4$ will dominate, and since it is still negative, $h(x)$ will again approach $-\mathrm{\infty }$.

Conclusion

Thus, the end behavior of the polynomial function $h(x)=-3x^2(5x+5)(x-1)$ is as follows:

• As $x\to +\mathrm{\infty }$, $h(x)\to -\mathrm{\infty }$.
• As $x\to -\mathrm{\infty }$, $h(x)\to -\mathrm{\infty }$.

The overall graphical representation of this polynomial will show that both ends of the graph approach $-\mathrm{\infty }$, creating a downward-facing parabola-like shape near the extremes.