Answer :
To analyze the behavior of the polynomial function given its roots and multiplicities, as well as the conditions of a positive leading coefficient and an even degree, we follow these steps:
1. Construct the Polynomial Function:
The roots and their multiplicities allow us to write the polynomial function as:
[tex]\[ (x + 7)^2 (x + 1) (x - 2)^4 (x - 4) \][/tex]
2. Behavior at Infinity:
Since the degree of the polynomial is even and the leading coefficient is positive, the graph of the polynomial will approach positive infinity as [tex]\( x \)[/tex] approaches both positive and negative infinity.
- For [tex]\( x \to -\infty \)[/tex], the graph goes to [tex]\( +\infty \)[/tex].
- For [tex]\( x \to \infty \)[/tex], the graph goes to [tex]\( +\infty \)[/tex].
3. Behavior Around Roots:
To determine the behavior of the polynomial around its roots, consider the multiplicity of each root:
- Root at [tex]\( x = -7 \)[/tex] with multiplicity 2: The graph touches the x-axis and remains above or below it without crossing.
- Root at [tex]\( x = -1 \)[/tex] with multiplicity 1: The graph crosses the x-axis.
- Root at [tex]\( x = 2 \)[/tex] with multiplicity 4: The graph touches the x-axis and remains on the same side.
- Root at [tex]\( x = 4 \)[/tex] with multiplicity 1: The graph crosses the x-axis.
4. Sign Analysis Around Intervals:
- Interval [tex]\( (-\infty, -7) \)[/tex]: The polynomial starts from [tex]\( +\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex] and remains positive as the [tex]\( (x + 7)^2 \)[/tex] term does not change sign.
Therefore, the graph is positive on [tex]\( (-\infty, -7) \)[/tex].
- Interval [tex]\( (-7, -1) \)[/tex]: The polynomial touches the x-axis at [tex]\( x = -7 \)[/tex] and changes sign at [tex]\( x = -1 \)[/tex]. It becomes negative after crossing [tex]\( x = -1 \)[/tex].
Therefore, the graph is negative on [tex]\( (-7, -1) \)[/tex].
- Interval [tex]\( (2, 4) \)[/tex]: At both ends, the polynomial [tex]\( (x - 2)^4 \)[/tex] term ensures it remains non-negative and positive overall.
Therefore, the graph is positive on [tex]\( (2, 4) \)[/tex].
- Interval [tex]\( (4, \infty) \)[/tex]: The polynomial crosses the x-axis at [tex]\( x = 4 \)[/tex] and tends towards [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] increases.
Therefore, the graph cannot be negative on [tex]\( (4, \infty) \)[/tex].
In conclusion, given the information about the polynomial, the statement that is true about the behavior of the graph is:
The graph of the function is negative on [tex]\((-7, -1)\)[/tex].
1. Construct the Polynomial Function:
The roots and their multiplicities allow us to write the polynomial function as:
[tex]\[ (x + 7)^2 (x + 1) (x - 2)^4 (x - 4) \][/tex]
2. Behavior at Infinity:
Since the degree of the polynomial is even and the leading coefficient is positive, the graph of the polynomial will approach positive infinity as [tex]\( x \)[/tex] approaches both positive and negative infinity.
- For [tex]\( x \to -\infty \)[/tex], the graph goes to [tex]\( +\infty \)[/tex].
- For [tex]\( x \to \infty \)[/tex], the graph goes to [tex]\( +\infty \)[/tex].
3. Behavior Around Roots:
To determine the behavior of the polynomial around its roots, consider the multiplicity of each root:
- Root at [tex]\( x = -7 \)[/tex] with multiplicity 2: The graph touches the x-axis and remains above or below it without crossing.
- Root at [tex]\( x = -1 \)[/tex] with multiplicity 1: The graph crosses the x-axis.
- Root at [tex]\( x = 2 \)[/tex] with multiplicity 4: The graph touches the x-axis and remains on the same side.
- Root at [tex]\( x = 4 \)[/tex] with multiplicity 1: The graph crosses the x-axis.
4. Sign Analysis Around Intervals:
- Interval [tex]\( (-\infty, -7) \)[/tex]: The polynomial starts from [tex]\( +\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex] and remains positive as the [tex]\( (x + 7)^2 \)[/tex] term does not change sign.
Therefore, the graph is positive on [tex]\( (-\infty, -7) \)[/tex].
- Interval [tex]\( (-7, -1) \)[/tex]: The polynomial touches the x-axis at [tex]\( x = -7 \)[/tex] and changes sign at [tex]\( x = -1 \)[/tex]. It becomes negative after crossing [tex]\( x = -1 \)[/tex].
Therefore, the graph is negative on [tex]\( (-7, -1) \)[/tex].
- Interval [tex]\( (2, 4) \)[/tex]: At both ends, the polynomial [tex]\( (x - 2)^4 \)[/tex] term ensures it remains non-negative and positive overall.
Therefore, the graph is positive on [tex]\( (2, 4) \)[/tex].
- Interval [tex]\( (4, \infty) \)[/tex]: The polynomial crosses the x-axis at [tex]\( x = 4 \)[/tex] and tends towards [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] increases.
Therefore, the graph cannot be negative on [tex]\( (4, \infty) \)[/tex].
In conclusion, given the information about the polynomial, the statement that is true about the behavior of the graph is:
The graph of the function is negative on [tex]\((-7, -1)\)[/tex].
