To determine the polynomial function \( f(x) \) with the specified characteristics, let’s go through the analysis step-by-step:
### Step 1: Identifying the Roots and Their Multiplicities
Given the roots and their multiplicities:
- Root at \( x = -4 \) with multiplicity 1
- Root at \( x = 5 \) with multiplicity 1
- Root at \( x = 1 \) with multiplicity 2
The general form of \( f(x) \) can be written as:
[tex]\[ f(x) = a(x + 4)(x - 5)(x - 1)^2 \][/tex]
where \( a \) is a constant multiplier that needs to be determined.
### Step 2: Utilizing the Y-Intercept
We know that the y-intercept of the function occurs when \( x = 0 \).
Given that the y-intercept is 40, we have:
[tex]\[ f(0) = 40 \][/tex]
### Step 3: Substitute \( x = 0 \) into the Polynomial
[tex]\[ f(0) = a(0 + 4)(0 - 5)(0 - 1)^2 \][/tex]
[tex]\[ f(0) = a \cdot 4 \cdot (-5) \cdot 1 \][/tex]
[tex]\[ 40 = a \cdot 4 \cdot (-5) \][/tex]
[tex]\[ 40 = a \cdot (-20) \][/tex]
[tex]\[ a = \frac{40}{-20} \][/tex]
[tex]\[ a = -2 \][/tex]
### Step 4: Construct the Polynomial with the Determined Multiplier
Now, substitute \( a = -2 \) back into the polynomial:
[tex]\[ f(x) = -2(x + 4)(x - 5)(x - 1)^2 \][/tex]
### Step 5: Expand the Polynomial Expression
Expand the expression to find the polynomial in standard form:
[tex]\[ f(x) = -2(x + 4)(x - 5)(x - 1)^2 \][/tex]
First, expand \( (x - 1)^2 \):
[tex]\[ (x - 1)^2 = x^2 - 2x + 1 \][/tex]
Next, expand \( (x + 4)(x - 5) \):
[tex]\[ (x + 4)(x - 5) = x^2 - x - 20 \][/tex]
Now, multiply the expanded terms:
[tex]\[ f(x) = -2(x^2 - x - 20)(x^2 - 2x + 1) \][/tex]
Further expanding:
[tex]\[ (x^2 - x - 20)(x^2 - 2x + 1) = x^4 - 2x^3 + x^2 - x^3 + 2x^2 - x - 20x^2 + 40x - 20 \][/tex]
[tex]\[ = x^4 - 3x^3 - 17x^2 + 39x - 20 \][/tex]
Multiply by \(-2\):
[tex]\[ f(x) = -2(x^4 - 3x^3 - 17x^2 + 39x - 20) \][/tex]
[tex]\[ = -2x^4 + 6x^3 + 34x^2 - 78x + 40 \][/tex]
Thus, the polynomial function is:
[tex]\[ f(x) = -2x^4 + 6x^3 + 34x^2 - 78x + 40 \][/tex]
