Question
Describe the end behavior of the polynomial
A. Down down
B. Up up
C. Up down
D. Down up
Polynomial: x^3 + 9x^2 + 23x + 15
A. Down down
B. Up up
C. Up down
D. Down up
Polynomial: x^3 + 9x^2 + 23x + 15
Answers
Answered by
GPT-4o mini
To determine the end behavior of a polynomial, we need to look at the leading term, which is the term with the highest degree. For the polynomial given:
\( f(x) = x^3 + 9x^2 + 23x + 15 \)
The leading term is \( x^3 \), which has an odd degree (3).
The end behavior of a polynomial is influenced by both the degree and the leading coefficient. In this case:
- Because the degree (3) is odd, the ends of the polynomial will go in opposite directions.
- The leading coefficient is positive (1), indicating that as \( x \) approaches positive infinity, \( f(x) \) will approach positive infinity, and as \( x \) approaches negative infinity, \( f(x) \) will go to negative infinity.
Thus, the end behavior is:
- As \( x \to +\infty, f(x) \to +\infty \) (up)
- As \( x \to -\infty, f(x) \to -\infty \) (down)
This corresponds to option **C: Up down**.
\( f(x) = x^3 + 9x^2 + 23x + 15 \)
The leading term is \( x^3 \), which has an odd degree (3).
The end behavior of a polynomial is influenced by both the degree and the leading coefficient. In this case:
- Because the degree (3) is odd, the ends of the polynomial will go in opposite directions.
- The leading coefficient is positive (1), indicating that as \( x \) approaches positive infinity, \( f(x) \) will approach positive infinity, and as \( x \) approaches negative infinity, \( f(x) \) will go to negative infinity.
Thus, the end behavior is:
- As \( x \to +\infty, f(x) \to +\infty \) (up)
- As \( x \to -\infty, f(x) \to -\infty \) (down)
This corresponds to option **C: Up down**.