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).
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.
To describe and explain the end behavior of a polynomial function, we need to look at the leading term of the polynomial.
In the given polynomial function h(x) = -3x^2 (5x + 5)(x - 1), the leading term is -3x^2.
Since the leading term is -3x^2, we can determine the end behavior based on the degree and the sign of the leading coefficient.
Firstly, the degree of the polynomial is 2 because of the highest power of x is 2.
Secondly, the leading coefficient is -3.
Now, let's consider the two possibilities for the end behavior based on the value of the leading coefficient:
1. If the leading coefficient is positive (+), the end behavior will be as follows:
- As x approaches negative infinity, h(x) will go to positive infinity (+∞).
- As x approaches positive infinity, h(x) will also go to positive infinity (+∞).
2. If the leading coefficient is negative (-), the end behavior will be as follows:
- As x approaches negative infinity, h(x) will go to negative infinity (-∞).
- As x approaches positive infinity, h(x) will also go to negative infinity (-∞).
In this case, the leading coefficient is negative (-3), so the end behavior of h(x) will be as follows:
- As x approaches negative infinity, h(x) will go to negative infinity (-∞).
- As x approaches positive infinity, h(x) will also go to negative infinity (-∞).
The end behavior of the given polynomial function is that the graph of h(x) will decrease without bound as x approaches positive and negative infinity.
To describe and explain the end behavior of a polynomial function, we can look at the leading term, which is the term with the highest degree in the polynomial. In this case, the leading term is -3x^2.
The degree of -3x^2 is 2, which is even. When a polynomial has an even degree and a negative leading coefficient like -3, its end behavior is as follows:
As x approaches negative infinity (-∞), the leading term -3x^2 becomes very large negative numbers. Thus, the graph of the polynomial approaches negative infinity on the left side.
As x approaches positive infinity (+∞), the leading term -3x^2 becomes very large positive numbers. Thus, the graph of the polynomial approaches positive infinity on the right side.
In summary, the end behavior of the polynomial function h(x) = -3x^2 (5x + 5)(x - 1) is that as x goes to negative infinity, the graph approaches negative infinity, and as x goes to positive infinity, the graph approaches positive infinity.