To find a polynomial function with the given zeros [tex]\(-3\)[/tex], [tex]\(2\)[/tex], and [tex]\(5\)[/tex] that passes through the point [tex]\((6, 108)\)[/tex], follow these steps:
### Step 1: Form the Polynomial from the Zeros
First, construct the polynomial [tex]\( f(x) \)[/tex] by using its roots. The polynomial with roots (zeros) [tex]\(-3\)[/tex], [tex]\(2\)[/tex], and [tex]\(5\)[/tex] is formed by:
[tex]\[ f(x) = a(x + 3)(x - 2)(x - 5) \][/tex]
Here, [tex]\( a \)[/tex] is a constant that we need to determine.
### Step 2: Expand the Polynomial
Next, expand the factors:
[tex]\[ (x + 3)(x - 2)(x - 5) \][/tex]
Start with two of the factors:
[tex]\[ (x + 3)(x - 2) = x^2 + x - 6 \][/tex]
Now multiply this result by the remaining factor:
[tex]\[ (x^2 + x - 6)(x - 5) \][/tex]
Use the distributive property to expand:
[tex]\[
\begin{align*}
(x^2 + x - 6)(x - 5) &= x^2(x - 5) + x(x - 5) - 6(x - 5) \\
&= x^3 - 5x^2 + x^2 - 5x - 6x + 30 \\
&= x^3 - 4x^2 - 11x + 30
\end{align*}
\][/tex]
So, the polynomial becomes:
[tex]\[ f(x) = a(x^3 - 4x^2 - 11x + 30) \][/tex]
### Step 3: Determine the Constant [tex]\( a \)[/tex]
To find the value of [tex]\( a \)[/tex], use the condition that the graph passes through the point [tex]\( (6, 108) \)[/tex]. This means:
[tex]\[ f(6) = 108 \][/tex]
Substitute [tex]\( x = 6 \)[/tex] into the polynomial:
[tex]\[
f(6) = a(6^3 - 4 \cdot 6^2 - 11 \cdot 6 + 30)
\][/tex]
Calculate the expression inside the parentheses:
[tex]\[
\begin{align*}
6^3 &= 216 \\
4 \cdot 6^2 &= 4 \cdot 36 = 144 \\
11 \cdot 6 &= 66 \\
6^3 - 4 \cdot 6^2 - 11 \cdot 6 + 30 &= 216 - 144 - 66 + 30 \\
&= 216 - 210 \\
&= 6
\end{align*}
\][/tex]
Thus:
[tex]\[
108 = a \cdot 6 \Rightarrow a = \frac{108}{6} = 18
\][/tex]
Thus:
[tex]\[
f(x) = 18(x^3 - 4x^2 - 11x + 30)
\][/tex]
### Step 4: Verify and Simplify
Plugging [tex]\( a = 3 \)[/tex], not 18 into the factored polynomial [tex]\( f(6) = 108 \)[/tex]:
[tex]\[ f(x) = 3(x^3 - 4x^2 - 11x + 30) \][/tex]
So the simplified polynomial function is:
[tex]\[ f(x) = 3x^3 - 12x^2 - 33x + 90 \][/tex]
Therefore, the polynomial function with the given conditions is:
[tex]\[ f(x) = 3x^3 - 12x^2 - 33x + 90 \][/tex]
