Sure, let's find an expression for the cubic function [tex]\( f(x) \)[/tex] given the conditions [tex]\( f(5) = 100 \)[/tex] and [tex]\( f(-5) = f(0) = f(6) = 0 \)[/tex].
### Step-by-Step Solution
1. Identify the Roots:
Since [tex]\( f(-5) = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex], and [tex]\( f(6) = 0 \)[/tex], these points (-5, 0, 6) are the roots of the cubic function. Hence, we can express the cubic function as:
[tex]\[
f(x) = a \cdot (x - r_1) \cdot (x - r_2) \cdot (x - r_3)
\][/tex]
where [tex]\( r_1, r_2, \)[/tex] and [tex]\( r_3 \)[/tex] are the roots of the function.
2. Substitute the Roots:
Given the roots, we have [tex]\( r_1 = -5 \)[/tex], [tex]\( r_2 = 0 \)[/tex], and [tex]\( r_3 = 6 \)[/tex]. Thus, the cubic function becomes:
[tex]\[
f(x) = a \cdot (x + 5) \cdot x \cdot (x - 6)
\][/tex]
3. Determine the Coefficient [tex]\( a \)[/tex]:
We are given [tex]\( f(5) = 100 \)[/tex]. Using this information, we can solve for the coefficient [tex]\( a \)[/tex]:
[tex]\[
f(5) = a \cdot (5 + 5) \cdot 5 \cdot (5 - 6)
\][/tex]
Simplify the expression:
[tex]\[
100 = a \cdot 10 \cdot 5 \cdot (-1)
\][/tex]
[tex]\[
100 = -50a
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = -2
\][/tex]
4. Formulate the Final Function:
Now that we have the value of [tex]\( a \)[/tex], we can write the final cubic function as:
[tex]\[
f(x) = -2 \cdot (x + 5) \cdot x \cdot (x - 6)
\][/tex]
### Conclusion
The cubic function that satisfies the given conditions [tex]\( f(5) = 100 \)[/tex] and [tex]\( f(-5) = f(0) = f(6) = 0 \)[/tex] is:
[tex]\[
f(x) = -2 \cdot (x + 5) \cdot x \cdot (x - 6)
\][/tex]
This is the required cubic function.
