Find the degree, leading term, leading coefficient, constant term, and end behavior of the given polynomial. $$ g(x)=6+x+3 x^{5}-6 x^{7} $$ Degree: Leading term: Leading coefficient: End behavior: As $$ x \rightarrow-\infty, g(x) \rightarrow \square $$ As $$ x \rightarrow \infty, g(x) \rightarrow \square $$

Question

UpStudy AI · Accepted Answer

Let's analyze the polynomial $$ g(x) = 6 + x + 3x^{5} - 6x^{7} $$ to find its degree, leading term, leading coefficient, constant term, and end behavior.

### 1. Degree:
The **degree** of a polynomial is the highest power of $$ x $$ present in the polynomial.

- **Highest power of $$ x $$**: $$ 7 $$ (from the term $$ -6x^{7} $$)

**Degree:** $$ 7 $$

---

### 2. Leading Term:
The **leading term** is the term with the highest degree in the polynomial.

- **Leading term:** $$ -6x^{7} $$

**Leading Term:** $$ -6x^{7} $$

---

### 3. Leading Coefficient:
The **leading coefficient** is the coefficient of the leading term.

- **Coefficient of $$ -6x^{7} $$:** $$ -6 $$

**Leading Coefficient:** $$ -6 $$

---

### 4. Constant Term:
The **constant term** is the term in the polynomial that does not contain $$ x $$.

- **Constant term:** $$ 6 $$

**Constant Term:** $$ 6 $$

---

### 5. End Behavior:
The **end behavior** of a polynomial describes how $$ g(x) $$ behaves as $$ x $$ approaches positive and negative infinity. This behavior is primarily determined by the leading term.

- **Leading Term:** $$ -6x^{7} $$
- **Degree:** Odd ($$ 7 $$)
- **Leading Coefficient:** Negative ($$ -6 $$)

For polynomials with:
- **Odd degree and negative leading coefficient:**
  - As $$ x \rightarrow -\infty $$, $$ g(x) \rightarrow \infty $$
  - As $$ x \rightarrow \infty $$, $$ g(x) \rightarrow -\infty $$

**End Behavior:**
- As $$ x \rightarrow -\infty $$, $$ g(x) \rightarrow \infty $$
- As $$ x \rightarrow \infty $$, $$ g(x) \rightarrow -\infty $$

---

### Summary:

- **Degree:** $$ 7 $$
- **Leading Term:** $$ -6x^{7} $$
- **Leading Coefficient:** $$ -6 $$
- **Constant Term:** $$ 6 $$
- **End Behavior:**
  - As $$ x \rightarrow -\infty $$, $$ g(x) \rightarrow \infty $$
  - As $$ x \rightarrow \infty $$, $$ g(x) \rightarrow -\infty $$
Degree: 7 Leading Term: -6x⁷ Leading Coefficient: -6 Constant Term: 6 End Behavior: As $$ x \rightarrow -\infty $$, $$ g(x) \rightarrow \infty $$ As $$ x \rightarrow \infty $$, $$ g(x) \rightarrow -\infty $$

Solución de tutoría real

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Algebra Questions