Questions: The graph is shown for f(x) = (x^4 + 3x^3 - 39x^2 - 47x + 210) / (x^4 - 2x^3 - 21x^2 + 22x + 40) This function can be expressed in factored form as: f(x) = (x+3) * (x-5) * (x+7) * (x-2) / (x+4) * (x-5) * (x+1) * (x-2) Which of the following is a TRUE statement? There are discontinuities at x=-4, x=-1, x=2, and x=5, which cause the denominator to become zero. There are discontinuities at x=-4 and x=-1 only (nowhere else), which are shown as locations of vertical asymptotes. There are discontinuities at x=2 and x=5 only (nowhere else), which are shown as holes in the graph.

The graph is shown for
f(x) = (x^4 + 3x^3 - 39x^2 - 47x + 210) / (x^4 - 2x^3 - 21x^2 + 22x + 40)

This function can be expressed in factored form as:
f(x) = (x+3) * (x-5) * (x+7) * (x-2) / (x+4) * (x-5) * (x+1) * (x-2)

Which of the following is a TRUE statement?
There are discontinuities at x=-4, x=-1, x=2, and x=5, which cause the denominator to become zero.
There are discontinuities at x=-4 and x=-1 only (nowhere else), which are shown as locations of vertical asymptotes.
There are discontinuities at x=2 and x=5 only (nowhere else), which are shown as holes in the graph.

Transcript text: The graph is shown for \[ f(x)=\frac{\left(x^{\wedge} 4+3 x^{\wedge} 3-39 x^{\wedge} 2-47 x+210\right)}{\left(x^{\wedge} 4-2 x^{\wedge} 3-21 x^{\wedge} 2+22 x+40\right)} \] This function can be expressed in factored form as: \[ f(x)=\frac{(x+3) \cdot(x-5) \cdot(x+7) \cdot(x-2)}{(x+4) \cdot(x-5) \cdot(x+1) \cdot(x-2)} \] Which of the following is a TRUE statement? There are discontinuities at $x=-4, x=-1, x=2$, and $x=5$, which cause the denominator to become zero. There are discontinuities at $x=-4$ and $x=-1$ only (nowhere else), which are shown as locations of vertical asymptotes. There are discontinuities at $\mathrm{x}=2$ and $\mathrm{x}=5$ only (nowhere else), which are shown as holes in the graph.

Solution

Solution Steps

To determine the true statement about the discontinuities of the function, we need to analyze the factored form of the function. Discontinuities occur where the denominator is zero. If a factor is present in both the numerator and the denominator, it indicates a removable discontinuity (hole). If a factor is only in the denominator, it indicates a vertical asymptote.

Identify the roots of the denominator: $x = -4, x = -1, x = 2, x = 5$.
Check if these roots are also present in the numerator:
- $x = 2$ and $x = 5$ are present in both the numerator and the denominator, indicating holes.
- $x = -4$ and $x = -1$ are only in the denominator, indicating vertical asymptotes.

Step 1: Identify the Roots of the Denominator

The denominator of the function is given by $(x + 4)(x - 5)(x + 1)(x - 2)$. The roots of the denominator, which are the values of $x$ that make the denominator zero, are $x = -4$, $x = -1$, $x = 2$, and $x = 5$.

Step 2: Determine the Nature of Discontinuities

To determine the nature of the discontinuities, we need to check if these roots are also present in the numerator $(x + 3)(x - 5)(x + 7)(x - 2)$.

Removable Discontinuities (Holes): These occur where a factor is present in both the numerator and the denominator. The roots $x = 2$ and $x = 5$ are present in both, indicating holes at these points.
Vertical Asymptotes: These occur where a factor is only in the denominator. The roots $x = -4$ and $x = -1$ are only in the denominator, indicating vertical asymptotes at these points.

Final Answer

The true statement is: There are discontinuities at $x = 2$ and $x = 5$ only (nowhere else), which are shown as holes in the graph. The answer is the third statement.

\[ \boxed{\text{There are discontinuities at } x = 2 \text{ and } x = 5 \text{ only (nowhere else), which are shown as holes in the graph.}} \]

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