Use the Leading Coefficient Test to determine the end behavior of the polynomial function.
7)

Question

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. 7) $$ f(x)=-x^{2}(x+1)(x+3) $$

UpStudy AI · Accepted Answer

To determine the end behavior of the polynomial function $$ f(x) = -x^2(x+1)(x+3) $$ using the Leading Coefficient Test, we first need to identify the leading term of the polynomial.

1. **Expand the polynomial**: 
   We start by expanding $$ f(x) $$:
   $$
   f(x) = -x^2(x+1)(x+3)
   $$
   First, expand $$ (x+1)(x+3) $$:
   $$
   (x+1)(x+3) = x^2 + 4x + 3
   $$
   Now, substitute this back into $$ f(x) $$:
   $$
   f(x) = -x^2(x^2 + 4x + 3)
   $$
   Next, distribute $$ -x^2 $$:
   $$
   f(x) = -x^4 - 4x^3 - 3x^2
   $$

2. **Identify the leading term**:
   The leading term of the polynomial is $$ -x^4 $$.

3. **Determine the leading coefficient and degree**:
   - The leading coefficient is $$ -1 $$ (since it is the coefficient of $$ -x^4 $$).
   - The degree of the polynomial is $$ 4 $$ (the highest power of $$ x $$).

4. **Apply the Leading Coefficient Test**:
   The Leading Coefficient Test states:
   - If the degree of the polynomial is even and the leading coefficient is positive, then as $$ x \to \infty $$ and $$ x \to -\infty $$, $$ f(x) \to \infty $$.
   - If the degree of the polynomial is even and the leading coefficient is negative, then as $$ x \to \infty $$ and $$ x \to -\infty $$, $$ f(x) \to -\infty $$.
   - If the degree of the polynomial is odd and the leading coefficient is positive, then as $$ x \to \infty $$, $$ f(x) \to \infty $$ and as $$ x \to -\infty $$, $$ f(x) \to -\infty $$.
   - If the degree of the polynomial is odd and the leading coefficient is negative, then as $$ x \to \infty $$, $$ f(x) \to -\infty $$ and as $$ x \to -\infty $$, $$ f(x) \to \infty $$.

Since our polynomial $$ f(x) $$ has:
- An even degree (4)
- A negative leading coefficient (-1)

According to the Leading Coefficient Test, the end behavior of the polynomial function $$ f(x) $$ is:
$$
\text{As } x \to \infty, f(x) \to -\infty \quad \text{and} \quad \text{As } x \to -\infty, f(x) \to -\infty.
$$

Thus, the end behavior of the polynomial function $$ f(x) = -x^2(x+1)(x+3) $$ is:
$$
\text{Both ends go to } -\infty.
$$
The end behavior of $$ f(x) = -x^2(x+1)(x+3) $$ is both ends go to $$ -\infty $$.

Use the Leading Coefficient Test to determine the end behavior of the polynomial function.
7)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Pre Calculus Questions