Question
Refer to the polynomial function h(x) = -3x^2(5x+3)(x-2 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).
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GPT-4o mini
To analyze the end behavior of the polynomial function \( h(x) = -3x^2(5x + 3)(x - 2) \), we first need to determine the degree and leading coefficient of the polynomial.
### Step 1: Find the degree of the polynomial
The function can be expressed as the product of its factors. We have:
1. The term \( -3x^2 \), which has a degree of 2.
2. The term \( (5x + 3) \), which has a degree of 1.
3. The term \( (x - 2) \), which has a degree of 1.
To find the total degree of \( h(x) \), we add the degrees of all terms:
\[
\text{Degree} = 2 + 1 + 1 = 4
\]
### Step 2: Identify the leading coefficient
The leading term of the polynomial comes from multiplying the leading coefficients of each factor. The leading term from \( -3x^2 \) is \( -3x^2 \), and the leading term from the polynomial expression \( (5x + 3)(x - 2) \) is \( 5x^2 \) (since \( 5x \cdot x \)). Thus, the leading term of \( h(x) \) is:
\[
-3x^2 \cdot 5x^2 = -15x^4
\]
So, the leading coefficient is \( -15 \).
### Step 3: Determine end behavior based on degree and leading coefficient
Now that we have established that \( h(x) \) is a polynomial of degree 4 (even degree) with a leading coefficient of \( -15 \) (negative), we can analyze the end behavior:
- For polynomials with an **even degree**:
- As \( x \) approaches \( +\infty \) (right end), \( h(x) \) approaches \( -\infty \) (downward).
- As \( x \) approaches \( -\infty \) (left end), \( h(x) \) also approaches \( -\infty \) (downward).
### Conclusion
The end behavior of the polynomial function \( h(x) = -3x^2(5x + 3)(x - 2) \) is described as follows:
- As \( x \to +\infty, h(x) \to -\infty \)
- As \( x \to -\infty, h(x) \to -\infty \)
This means that both ends of the graph of \( h(x) \) will trend downwards as you move to the left or right.
### Step 1: Find the degree of the polynomial
The function can be expressed as the product of its factors. We have:
1. The term \( -3x^2 \), which has a degree of 2.
2. The term \( (5x + 3) \), which has a degree of 1.
3. The term \( (x - 2) \), which has a degree of 1.
To find the total degree of \( h(x) \), we add the degrees of all terms:
\[
\text{Degree} = 2 + 1 + 1 = 4
\]
### Step 2: Identify the leading coefficient
The leading term of the polynomial comes from multiplying the leading coefficients of each factor. The leading term from \( -3x^2 \) is \( -3x^2 \), and the leading term from the polynomial expression \( (5x + 3)(x - 2) \) is \( 5x^2 \) (since \( 5x \cdot x \)). Thus, the leading term of \( h(x) \) is:
\[
-3x^2 \cdot 5x^2 = -15x^4
\]
So, the leading coefficient is \( -15 \).
### Step 3: Determine end behavior based on degree and leading coefficient
Now that we have established that \( h(x) \) is a polynomial of degree 4 (even degree) with a leading coefficient of \( -15 \) (negative), we can analyze the end behavior:
- For polynomials with an **even degree**:
- As \( x \) approaches \( +\infty \) (right end), \( h(x) \) approaches \( -\infty \) (downward).
- As \( x \) approaches \( -\infty \) (left end), \( h(x) \) also approaches \( -\infty \) (downward).
### Conclusion
The end behavior of the polynomial function \( h(x) = -3x^2(5x + 3)(x - 2) \) is described as follows:
- As \( x \to +\infty, h(x) \to -\infty \)
- As \( x \to -\infty, h(x) \to -\infty \)
This means that both ends of the graph of \( h(x) \) will trend downwards as you move to the left or right.
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