Answer :
To find the equation of the quartic function given its roots and leading coefficient, let's go through the process step-by-step:
### Step 1: Identify the Roots
The given roots of the quartic function are:
- [tex]\( x = -3 \)[/tex]
- [tex]\( x = 0 \)[/tex]
- [tex]\( x = -5i \)[/tex]
Since complex roots come in conjugate pairs, the root [tex]\( x = -5i \)[/tex] implies there is also a root [tex]\( x = 5i \)[/tex].
### Step 2: Form Factorized Expression
The factorized form of the polynomial based on these roots is:
[tex]\[ (x + 3)(x)(x - 5i)(x + 5i) \][/tex]
### Step 3: Simplify Complex Roots
The expression [tex]\( (x - 5i)(x + 5i) \)[/tex] can be simplified using the difference of squares formula:
[tex]\[ (x - 5i)(x + 5i) = x^2 - (5i)^2 = x^2 - (-25) = x^2 + 25 \][/tex]
### Step 4: Expand the Polynomial
Now, substitute the simplified expression:
The polynomial becomes:
[tex]\[ (x + 3)(x)(x^2 + 25) \][/tex]
First, expand [tex]\( (x + 3)(x) \)[/tex]:
[tex]\[ (x + 3)x = x^2 + 3x \][/tex]
Now multiply this result by [tex]\( (x^2 + 25) \)[/tex]:
[tex]\[ (x^2 + 3x)(x^2 + 25) \][/tex]
Expanding these two binomials gives:
[tex]\[ x^2(x^2 + 25) + 3x(x^2 + 25) \][/tex]
Calculate each part:
- [tex]\( x^2 \cdot x^2 = x^4 \)[/tex]
- [tex]\( x^2 \cdot 25 = 25x^2 \)[/tex]
- [tex]\( 3x \cdot x^2 = 3x^3 \)[/tex]
- [tex]\( 3x \cdot 25 = 75x \)[/tex]
Putting it all together gives:
[tex]\[ x^4 + 3x^3 + 25x^2 + 75x \][/tex]
### Step 5: Apply Leading Coefficient
Since the leading coefficient is [tex]\(-1\)[/tex], multiply the entire polynomial by [tex]\(-1\)[/tex]:
[tex]\[ -1(x^4 + 3x^3 + 25x^2 + 75x) \][/tex]
This results in:
[tex]\[ -x^4 - 3x^3 - 25x^2 - 75x \][/tex]
Therefore, the equation of the quartic function is:
[tex]\[ -x^4 - 3x^3 - 25x^2 - 75x \][/tex]
This corresponds to the option:
[tex]\[ -x^4 - 3x^3 - 25x^2 - 75x \][/tex]
### Step 1: Identify the Roots
The given roots of the quartic function are:
- [tex]\( x = -3 \)[/tex]
- [tex]\( x = 0 \)[/tex]
- [tex]\( x = -5i \)[/tex]
Since complex roots come in conjugate pairs, the root [tex]\( x = -5i \)[/tex] implies there is also a root [tex]\( x = 5i \)[/tex].
### Step 2: Form Factorized Expression
The factorized form of the polynomial based on these roots is:
[tex]\[ (x + 3)(x)(x - 5i)(x + 5i) \][/tex]
### Step 3: Simplify Complex Roots
The expression [tex]\( (x - 5i)(x + 5i) \)[/tex] can be simplified using the difference of squares formula:
[tex]\[ (x - 5i)(x + 5i) = x^2 - (5i)^2 = x^2 - (-25) = x^2 + 25 \][/tex]
### Step 4: Expand the Polynomial
Now, substitute the simplified expression:
The polynomial becomes:
[tex]\[ (x + 3)(x)(x^2 + 25) \][/tex]
First, expand [tex]\( (x + 3)(x) \)[/tex]:
[tex]\[ (x + 3)x = x^2 + 3x \][/tex]
Now multiply this result by [tex]\( (x^2 + 25) \)[/tex]:
[tex]\[ (x^2 + 3x)(x^2 + 25) \][/tex]
Expanding these two binomials gives:
[tex]\[ x^2(x^2 + 25) + 3x(x^2 + 25) \][/tex]
Calculate each part:
- [tex]\( x^2 \cdot x^2 = x^4 \)[/tex]
- [tex]\( x^2 \cdot 25 = 25x^2 \)[/tex]
- [tex]\( 3x \cdot x^2 = 3x^3 \)[/tex]
- [tex]\( 3x \cdot 25 = 75x \)[/tex]
Putting it all together gives:
[tex]\[ x^4 + 3x^3 + 25x^2 + 75x \][/tex]
### Step 5: Apply Leading Coefficient
Since the leading coefficient is [tex]\(-1\)[/tex], multiply the entire polynomial by [tex]\(-1\)[/tex]:
[tex]\[ -1(x^4 + 3x^3 + 25x^2 + 75x) \][/tex]
This results in:
[tex]\[ -x^4 - 3x^3 - 25x^2 - 75x \][/tex]
Therefore, the equation of the quartic function is:
[tex]\[ -x^4 - 3x^3 - 25x^2 - 75x \][/tex]
This corresponds to the option:
[tex]\[ -x^4 - 3x^3 - 25x^2 - 75x \][/tex]
