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 \( x \to +\infty \):

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

2. As \( x \to -\infty \):

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

Conclusion

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

• As \( x \to +\infty \), \( h(x) \to -\infty \).
• As \( x \to -\infty \), \( h(x) \to -\infty \).

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