Sure! Let's build your polynomial step-by-step according to the given instructions:
1. Identifying Root Intercepts:
- We need to determine the factors of the polynomial based on the x-intercepts.
- We know the coaster should have an x-intercept at [tex]\( x = 500 \)[/tex], an intercept at [tex]\( x=0 \)[/tex], and another intercept at [tex]\( x=1000 \)[/tex].
2. Forming Factors:
- The x-intercepts [tex]\( x = 0 \)[/tex], [tex]\( x = 500 \)[/tex], and [tex]\( x = 1000 \)[/tex] will translate to polynomial factors. Remember, if the polynomial has an x-intercept at [tex]\( x = c \)[/tex], then [tex]\( (x - c) \)[/tex] will be a factor of the polynomial.
So the factors are:
- [tex]\( x \)[/tex] for the intercept at [tex]\( x = 0 \)[/tex]
- [tex]\( (x - 500) \)[/tex] for the intercept at [tex]\( x = 500 \)[/tex]
- [tex]\( (x - 1000) \)[/tex] for the intercept at [tex]\( x = 1000 \)[/tex]
3. Constructing the Polynomial:
- Multiply these factors together to form the polynomial.
- Since the polynomial needs to rise to a maximum, fall, and then rise again, we need the coefficients to capture this behavior adequately. The polynomial can be written as:
[tex]\[
y = a \cdot x \cdot (x - 500) \cdot (x - 1000)
\][/tex]
Where `a` is a constant coefficient which can be any real number.
4. General Form of the Polynomial:
- The final polynomial in the required form would therefore be:
[tex]\[
y = a \cdot x \cdot (x - 500) \cdot (x - 1000)
\][/tex]
This polynomial has the desired x-intercepts at [tex]\( x=0 \)[/tex], [tex]\( x=500 \)[/tex], and [tex]\( x=1000 \)[/tex], and shows the behavior mentioned in the initial description where it rises to a maximum, falls across the x-axis, and then rises again into [tex]\( x = 1000 \)[/tex].
