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Asymptotes are lines that a graph approaches but does not touch. They help us understand the behavior of functions at infinity or near undefined points. To find vertical asymptotes, we set the denominator equal to zero and solve for the variable. For the function \(f(x) = \frac{x+2}{(x-1)^3}\), setting \((x-1)^3 = 0\) gives us \(x = 1\). Hence, the vertical asymptote is at \(x = 1\).

There are no horizontal or oblique asymptotes in this case. Horizontal asymptotes usually occur when the degrees of the numerator and denominator are equal or when the degree of the numerator is less than the degree of the denominator. Since the degrees do not satisfy these conditions, we have no such asymptotes here.

Understanding asymptotes is crucial in graphing because they indicate where the function will trend toward positive or negative infinity.

Intercepts are points on a graph where the function crosses the axes. They give us specific, known points to help us sketch the graph accurately. To find the \(x\)-intercept, set the numerator of the function to zero and solve for \(x\). For \(f(x) = \frac{x+2}{(x-1)^3}\), setting \(x+2 = 0\) gives us \(x = -2\). Thus, the \(x\)-intercept is at \((-2, 0)\).

For the \(y\)-intercept, substitute \(x = 0\) into the function. So, \(f(0) = \frac{0+2}{(0-1)^3} = \frac{2}{-1} = -2\). Therefore, the \(y\)-intercept is at \((0, -2)\).

Intercepts are essential for drawing the graph as they provide clear points where the graph touches or crosses the axes.

The domain of a function includes all the values of \(x\) for which the function is defined. For rational functions, the domain excludes values that make the denominator zero because division by zero is undefined.

In the function \(f(x) = \frac{x+2}{(x-1)^3}\), the denominator \((x-1)^3\) equals zero when \(x = 1\). Therefore, \(x = 1\) cannot be included in the domain. The domain is thus all real numbers except \(x = 1\), written as \((-\infty, 1) \cup (1, \infty)\).

Knowing the domain is important as it tells us where the function is valid and can be graphically represented.

Polynomial division is a method for dividing polynomials, similar to long division with numbers. It helps us find slant asymptotes or to simplify complex rational functions.

For the function \(f(x) = \frac{x+2}{(x-1)^3}\), since the degree of the numerator (1) is less than the degree of the denominator (3), there is no need for polynomial division in this case. However, if you ever need to perform polynomial division, here’s a quick summary on how to do it:

• Divide the first term of the numerator by the first term of the denominator.
• Multiply the result by the entire denominator and subtract it from the original numerator.
• Continue this process until you can't divide anymore.

This is useful to find the function's behavior at large values of \(x\) or to identify slant (oblique) asymptotes.

While we didn't need polynomial division for this function, it's a valuable tool for more complex rational functions.