Question

end behavior of polynomials
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consider the polynomial function $g(x) = 10x^9 - 50x^6 - 500x^2$.
what is the end behavior of the graph of $g$?

Explanation:

Step1: Identify leading term

The leading term of $g(x)$ is $10x^9$.

Step2: Analyze degree parity

The degree of the polynomial is 9, which is odd.

Step3: Analyze leading coefficient sign

The leading coefficient is 10, which is positive.

Step4: Define end behavior rules

For odd degree with positive leading coefficient:

• As $x \to +\infty$, $g(x) \to +\infty$
• As $x \to -\infty$, $g(x) \to -\infty$

Answer:

As $x$ approaches positive infinity, $g(x)$ approaches positive infinity; as $x$ approaches negative infinity, $g(x)$ approaches negative infinity.
In notation:
$\lim_{x \to +\infty} g(x) = +\infty$
$\lim_{x \to -\infty} g(x) = -\infty$