Question

state the degree and end behavior of $f(x)=8x^{3}+2x^{4}-5x^{2}+9$. $f(x)$ has a degree of 4. as $x$ gets larger and larger in the positive direction, $f(x)$ gets larger and larger in the positive direction. as $x$ gets larger and larger in the negative direction, $f(x)$ gets larger and larger in the negative direction.

Explanation:

Step1: Identify the degree

The degree of a polynomial is the highest - power of the variable. In the polynomial $f(x)=8x^{3}+2x^{4}-5x^{2}+9$, the highest - power of $x$ is 4.

Step2: Analyze end - behavior for positive $x$

For a polynomial $f(x)=a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_0$ with $n$ (degree) even and $a_n>0$ (here $n = 4$ and $a_4=2>0$), as $x\to+\infty$, $f(x)\to+\infty$.

Step3: Analyze end - behavior for negative $x$

Since the degree $n = 4$ (even) and $a_n = 2>0$, when $x\to-\infty$, $f(x)=2(-x)^4+8(-x)^3-5(-x)^2 + 9=2x^4-8x^3 - 5x^2+9$. As $x$ is large negative, the leading term $2x^4$ dominates and $f(x)\to+\infty$. The original answer for the negative - direction end - behavior is incorrect.

Answer:

$f(x)$ has a degree of 4. As $x$ gets larger and larger in the positive direction, $f(x)$ gets larger and larger in the positive direction. As $x$ gets larger and larger in the negative direction, $f(x)$ gets larger and larger in the positive direction.