Question
Rogelio is asked to sketch a graph of g(x)=3x^3(x-5)^2(5-x)^4
Exactly eight
Eight at most
Nine at most
Three at most
Exactly eight
Eight at most
Nine at most
Three at most
Answers
GPT 3.5
To determine the number of x-intercepts, we need to find the values of x for which g(x) equals zero.
The function g(x) can be written as the product of three factors: 3x^3, (x-5)^2, and (5-x)^4.
1) The factor 3x^3 has three solutions: x = 0, x = 0, and x = 0 (triple root). However, a triple root is considered as one x-intercept.
2) The factor (x-5)^2 has two solutions: x = 5 and x = 5. Again, a double root is counted as one x-intercept.
3) The factor (5-x)^4 also has four solutions: x = 5, x = 5, x = 5, and x = 5. A quadruple root is still considered as one x-intercept.
Therefore, there are eight x-intercepts in total.
Answer: Exactly eight x-intercepts.
Note: The terms "exactly eight," "eight at most," "nine at most," and "three at most" refer to the maximum possible number of x-intercepts for the given function.
The function g(x) can be written as the product of three factors: 3x^3, (x-5)^2, and (5-x)^4.
1) The factor 3x^3 has three solutions: x = 0, x = 0, and x = 0 (triple root). However, a triple root is considered as one x-intercept.
2) The factor (x-5)^2 has two solutions: x = 5 and x = 5. Again, a double root is counted as one x-intercept.
3) The factor (5-x)^4 also has four solutions: x = 5, x = 5, x = 5, and x = 5. A quadruple root is still considered as one x-intercept.
Therefore, there are eight x-intercepts in total.
Answer: Exactly eight x-intercepts.
Note: The terms "exactly eight," "eight at most," "nine at most," and "three at most" refer to the maximum possible number of x-intercepts for the given function.