Question
Rogelio is asked to sketch a graph of g(x)=3x3(x−5)2(5−x)4 . How many turning points should his sketch include?(1 point)Responsesnine at mostnine at mosteight at mosteight at mostexactly eightexactly eightthree at most
Answers
GPT-4o mini
To determine the number of turning points of the function \( g(x) = 3x^3(x−5)^2(5−x)^4 \), we first analyze its components and behavior.
1. **Identify the polynomial degree**: The expression can be rewritten as \( g(x) = 3x^3(5 - x)^4(x - 5)^2 \). The degree can be calculated by summing the degrees of all terms:
- The degree of \( 3x^3 \) is 3.
- The degree of \( (x - 5)^2 \) is 2.
- The degree of \( (5 - x)^4 = -(x - 5)^4 \) is 4.
Therefore, the total degree is \( 3 + 2 + 4 = 9 \).
2. **Finding turning points**: The maximum number of turning points in a polynomial is given by the degree of the polynomial minus one. Thus, for \( g(x) \), the maximum number of turning points is \( 9 - 1 = 8 \).
3. **Multiplicity of roots**: Next, we consider the multiplicity of the roots:
- At \( x = 0 \), the root has a multiplicity of 3.
- At \( x = 5 \), the root has a multiplicity of 2.
- At \( x = 5 \) (considering \( 5-x \)), there is also a multiplicity of 4.
Critical points based on these roots must reduce the turning points depending on whether the roots are odd (causing a change in direction) or even (not causing a change in direction):
- \( x = 0 \) (odd multiplicity of 3) contributes 1 turning point.
- \( x = 5 \) (even multiplicity of 2) does not contribute turning points.
- \( x = 5 \) (even multiplicity of 4) also does not contribute turning points.
Based on this analysis, the total number of turning points will typically be fewer than the maximum due to the nature of the roots.
All of this suggests that while the maximum number of turning points is 8, because of the repeated roots and their multiplicities, the actual number may be less but capped at 8.
Thus, Rogelio’s sketch should include at most **eight turning points**.
The best choice is: **eight at most**.
1. **Identify the polynomial degree**: The expression can be rewritten as \( g(x) = 3x^3(5 - x)^4(x - 5)^2 \). The degree can be calculated by summing the degrees of all terms:
- The degree of \( 3x^3 \) is 3.
- The degree of \( (x - 5)^2 \) is 2.
- The degree of \( (5 - x)^4 = -(x - 5)^4 \) is 4.
Therefore, the total degree is \( 3 + 2 + 4 = 9 \).
2. **Finding turning points**: The maximum number of turning points in a polynomial is given by the degree of the polynomial minus one. Thus, for \( g(x) \), the maximum number of turning points is \( 9 - 1 = 8 \).
3. **Multiplicity of roots**: Next, we consider the multiplicity of the roots:
- At \( x = 0 \), the root has a multiplicity of 3.
- At \( x = 5 \), the root has a multiplicity of 2.
- At \( x = 5 \) (considering \( 5-x \)), there is also a multiplicity of 4.
Critical points based on these roots must reduce the turning points depending on whether the roots are odd (causing a change in direction) or even (not causing a change in direction):
- \( x = 0 \) (odd multiplicity of 3) contributes 1 turning point.
- \( x = 5 \) (even multiplicity of 2) does not contribute turning points.
- \( x = 5 \) (even multiplicity of 4) also does not contribute turning points.
Based on this analysis, the total number of turning points will typically be fewer than the maximum due to the nature of the roots.
All of this suggests that while the maximum number of turning points is 8, because of the repeated roots and their multiplicities, the actual number may be less but capped at 8.
Thus, Rogelio’s sketch should include at most **eight turning points**.
The best choice is: **eight at most**.