In the polynomial function f(x) = 4x(x-4)(x+2), what is the end behavior?

Understand the Problem

The question is asking us to determine the end behavior of the polynomial function $f(x) = 4x(x-4)(x+2)$. To solve this, we need to analyze the leading term of the polynomial and how it behaves as $x$ approaches positive and negative infinity.

Answer

As $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to -\infty$.

Steps to Solve

Identify the degree of the polynomial

First, we need to determine the degree of the polynomial. The polynomial $f(x) = 4x(x-4)(x+2)$ can be expanded. The degree is determined by the highest exponent of $x$ in the polynomial.

Expand the polynomial

We will expand the polynomial to find the leading term. First, expand the factors:

$$ f(x) = 4x[(x-4)(x+2)] $$

Now, expand $(x-4)(x+2)$:

$$ (x-4)(x+2) = x^2 + 2x - 4x - 8 = x^2 - 2x - 8 $$

Now substituting back into the polynomial:

$$ f(x) = 4x(x^2 - 2x - 8) $$

Next, we need to continue expanding.

Further expand and combine like terms

Now, distribute $4x$:

$$ f(x) = 4x^3 - 8x^2 - 32x $$

Identify the leading term

From our expansion, we see that the leading term is $4x^3$.

Analyze the leading term for end behavior

The leading term $4x^3$ will dominate the polynomial as $x$ approaches positive and negative infinity.

As $x \to \infty$, $f(x) \to \infty$ (since $4x^3 \to \infty$)
As $x \to -\infty$, $f(x) \to -\infty$ (since $4(-\infty)^3 \to -\infty$)

The end behavior of the polynomial function $f(x) = 4x(x-4)(x+2)$ is:

As $x \to \infty$, $f(x) \to \infty$
As $x \to -\infty$, $f(x) \to -\infty$

More Information

The leading term determines the end behavior of a polynomial function. In this case, since the leading term is $4x^3$, which is an odd-degree polynomial, the function will rise to infinity as $x$ increases and fall to negative infinity as $x$ decreases.

Tips

Forgetting to account for the leading coefficient when determining end behavior.
Confusing the behavior of even-degree and odd-degree polynomials.

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