Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
