Find the degree, leading term, leading coefficient, constant term, and end behavior of the
given polynomial.

Degree:
Leading term:
Leading coefficient:
End behavior:
As
As

Ask by Weston Wood. in the United States

Jan 23,2025

Question

Find the degree, leading term, leading coefficient, constant term, and end behavior of the
given polynomial.

Degree:
Leading term:
Leading coefficient:
End behavior:
As
As

Ask by Weston Wood. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

Let's analyze the polynomial $$ p(t) = t^{5}(2 - 5t)(t^{2} + t + 8) $$ to determine its degree, leading term, leading coefficient, constant term, and end behavior.

### 1. Degree
The degree of a polynomial is the highest power of the variable $$ t $$ after expanding the polynomial.

- $$ t^5 $$ has a degree of **5**.
- $$ (2 - 5t) $$ has a degree of **1**.
- $$ (t^2 + t + 8) $$ has a degree of **2**.

**Total Degree**: $$ 5 + 1 + 2 = 8 $$.

**Degree:** 8

### 2. Leading Term and Leading Coefficient
The leading term is found by multiplying the leading terms of each factor:

- Leading term of $$ t^5 $$ is $$ t^5 $$.
- Leading term of $$ (2 - 5t) $$ is $$ -5t $$.
- Leading term of $$ (t^2 + t + 8) $$ is $$ t^2 $$.

**Leading Term Calculation:**
$$
t^5 \times (-5t) \times t^2 = -5t^{8}
$$

**Leading Term:** $$ -5t^8 $$

The **leading coefficient** is the coefficient of the leading term, which is **-5**.

**Leading Coefficient:** -5

### 3. Constant Term
The constant term is the term in the polynomial that does not contain the variable $$ t $$.

When expanding $$ p(t) $$, every term will have a power of $$ t $$ higher than or equal to 5. Therefore, there is no constant (degree 0) term in this polynomial.

**Constant Term:** 0

### 4. End Behavior
The end behavior of a polynomial is determined by its leading term.

- **Leading Term:** $$ -5t^8 $$
  - Since the degree is even and the leading coefficient is negative, as $$ t $$ approaches both $$ \infty $$ and $$ -\infty $$, the polynomial will approach $$ -\infty $$.

**End Behavior:**
- As $$ t \rightarrow -\infty, \ p(t) \rightarrow -\infty $$
- As $$ t \rightarrow \infty, \ p(t) \rightarrow -\infty $$

### Summary

- **Degree:** 8
- **Leading term:** $$ -5t^8 $$
- **Leading coefficient:** -5
- **Constant term:** 0
- **End behavior:**
  - As $$ t \rightarrow -\infty, \ p(t) \rightarrow -\infty $$
  - As $$ t \rightarrow \infty, \ p(t) \rightarrow -\infty $$
**Degree:** 8 **Leading term:** $$ -5t^8 $$ **Leading coefficient:** -5 **Constant term:** 0 **End behavior:** - As $$ t \rightarrow -\infty, \ p(t) \rightarrow -\infty $$ - As $$ t \rightarrow \infty, \ p(t) \rightarrow -\infty $$

Find the degree, leading term, leading coefficient, constant term, and end behavior of the
given polynomial.

Degree:
Leading term:
Leading coefficient:
End behavior:
As
As

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Find the degree, leading term, leading coefficient, constant term, and end behavior of the given polynomial. Degree: Leading term: Leading coefficient: End behavior: As As

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Find the degree, leading term, leading coefficient, constant term, and end behavior of the
given polynomial.

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Leading term:
Leading coefficient:
End behavior:
As
As