The given polynomial is:
p(t) = -t²(3 - 5t)(t² + t + 4)
To find the degree of the polynomial, we need to determine the highest power of t in the expression. This is given by:
degree = 2 + 1 + 2 = 5
So the degree of the polynomial is 5.
The leading term of the polynomial is the term with the highest power of t. This is given by:
leading term = -t² * 5t² = -5t^4
So the leading term of the polynomial is -5t^4.
The leading coefficient of the polynomial is the coefficient of the leading term. This is given by:
leading coefficient = -5
So the leading coefficient of the polynomial is -5.
The constant term of the polynomial is the term that does not contain any powers of t. This is given by:
constant term = -t²(3)(4) = -12t²
So the constant term of the polynomial is -12t^2.
To find the end behavior of the polynomial, we need to determine what happens to the value of the polynomial as t approaches positive or negative infinity. Since the degree of the polynomial is odd, we know that the end behavior will be opposite for t approaching positive or negative infinity. We can use the leading term to determine the end behavior:
- as t approaches positive infinity, the leading term approaches negative infinity, so the end behavior is p(t) → -∞ as t → ∞
- as t approaches negative infinity, the leading term approaches positive infinity, so the end behavior is p(t) → ∞ as t → -∞
So the end behavior of the polynomial is:
- p(t) → -∞ as t → ∞
- p(t) → ∞ as t → -∞