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
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.
To sketch the graph of g(x) = 3x^3(x-5)^2(5-x)^4, we can start by identifying the key information about the function.
1. Number of x-intercepts (zeros): The function g(x) will have x-intercepts where g(x) = 0. We can find the zeros by setting each factor equal to zero and solving for x.
- The factor x^3 will have one zero at x = 0 since it is a third-degree polynomial.
- The factor (x-5)^2 will have one zero at x = 5 since it is a quadratic.
- The factor (5-x)^4 will have four zeros at x = 5 since it is a fourth-degree polynomial.
Therefore, g(x) will have eight x-intercepts: x = 0, x = 5 (repeated four times).
2. Behavior near x = 5: Since the factors (x-5)^2 and (5-x)^4 both involve (x-5), we can determine the behavior of g(x) near x = 5. The factors (x-5)^2 and (5-x)^4 indicate that g(x) will approach zero as x approaches 5 from both sides.
Now, let's answer the given questions step-by-step:
a) Exactly eight: The graph of g(x) will have exactly eight x-intercepts: x = 0, and x = 5 (repeated four times).
b) Eight at most: The graph of g(x) may have up to eight x-intercepts: x = 0, and x = 5 (repeated four times).
c) Nine at most: The graph of g(x) may have up to nine x-intercepts if it has an additional zero somewhere other than x = 0 and x = 5. However, according to the given function g(x), there are no additional factors or terms, so there can't be more than eight x-intercepts.
d) Three at most: The graph of g(x) may have up to three x-intercepts if it has zeros at x = 0 and two additional values. However, g(x) has four repeated zeros at x = 5, so there will be more than three x-intercepts.
Please note that this information provides an overview of the behavior of the function g(x) based on its factors. To get a more precise graph, you may want to use graphing software or plot the points manually using the calculated zeros.
To accurately sketch the graph of the function g(x) = 3x^3(x-5)^2(5-x)^4, let's analyze its key properties.
First, let's consider its degree. The degree of a polynomial is determined by the highest power of x in the equation. In this case, since the highest power of x is 4, the degree of g(x) is 4.
Next, let's look at the leading coefficient. The leading coefficient is the number multiplied by the variable with the highest power (in this case, x^4). The leading coefficient of g(x) is 3.
Now, we can determine the behavior of the function as x approaches positive and negative infinity:
- As x approaches positive infinity, the function g(x) will also approach positive infinity since the leading coefficient (3) and the highest-degree term (x^4) are both positive.
- Likewise, as x approaches negative infinity, the function g(x) will also approach positive infinity for the same reasons.
Now, we can find the x-intercepts (locations where the graph crosses the x-axis) for g(x). These correspond to the values of x that make g(x) equal to zero. From the given equation, we can see that x = 0, x = 5, and x = 5 (with multiplicity 4) are the zeros.
Based on this information, we can identify the number of x-intercepts the graph of g(x) will have:
- Exactly eight: Since there are three distinct zeros at x = 0 and x = 5, and the multiplicity at x = 5 is 4, the total number of x-intercepts is 3 + 4 = 7. However, we need to account for the possibility of another x-intercept between x = 0 and x = 5, which makes a total of exactly eight x-intercepts.
- Eight at most: This means that the number of x-intercepts can be any value up to eight, but not exceeding that.
- Nine at most: Similar to the previous option, this means the number of x-intercepts can be any value up to nine.
- Three at most: This indicates that the function can have a maximum of three x-intercepts.
To further analyze the behavior of the graph and accurately sketch it, we can also consider the sign of the function g(x) in different intervals between the zeros.
I hope this explanation helps you understand how to determine the number of x-intercepts for the given function g(x).