Find the degree, leading term, leading coefficient, constant term, and end behavior of the given polynomial. \( p(t)=t^{5}(2-5 t)\left(t^{2}+t+8\right) \) Degree: Leading term: Leading coefficient: End behavior: As \( t \rightarrow-\infty, p(t) \rightarrow \square \) As \( t \rightarrow \infty, p(t) \rightarrow \square \)

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 \( t \rightarrow \infty, \ p(t) \rightarrow -\infty \)