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To determine which polynomial function satisfies the condition of having a leading coefficient of 1 and roots [tex]\(2i\)[/tex] and [tex]\(3i\)[/tex] with multiplicity 1, we must look through the provided options and simplify each one to find its leading coefficient and its polynomial expression.
### Option 1: [tex]\(f(x) = (x - 2i)(x - 3i)\)[/tex]
Let's expand this:
[tex]\[(x - 2i)(x - 3i)\][/tex]
To expand, we apply the distributive property (also known as FOIL method):
[tex]\[ (x - 2i)(x - 3i) = x^2 - 3ix - 2ix + (2i \cdot 3i) \][/tex]
[tex]\[ = x^2 - 5ix + (-6) \][/tex]
[tex]\[ = x^2 - 5ix - 6 \][/tex]
### Option 2: [tex]\(f(x) = (x + 2i)(x + 3i)\)[/tex]
Let's expand this:
[tex]\[(x + 2i)(x + 3i)\][/tex]
Using the distributive property:
[tex]\[ (x + 2i)(x + 3i) = x^2 + 3ix + 2ix + (2i \cdot 3i) \][/tex]
[tex]\[ = x^2 + 5ix + (-6) \][/tex]
[tex]\[ = x^2 + 5ix - 6 \][/tex]
### Option 3: [tex]\(f(x) = (x - 2)(x - 3)(x - 2i)(x - 3i)\)[/tex]
Let's expand this step-by-step:
First, expand [tex]\((x - 2)(x - 3)\)[/tex]:
[tex]\[(x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6\][/tex]
Next, expand [tex]\((x - 2i)(x - 3i)\)[/tex] (we did this part in Option 1):
[tex]\[(x - 2i)(x - 3i) = x^2 - 5ix - 6\][/tex]
Now multiply [tex]\( (x^2 - 5x + 6) \)[/tex] by [tex]\((x^2 - 5ix - 6)\)[/tex]:
Without completing this entire expansion, observe that it introduces non-real coefficients and mix polynomial degrees, making it more complex than the desired result.
### Option 4: [tex]\(f(x) = (x + 2i)(x + 3i)(x - 2i)(x - 3i)\)[/tex]
Let's expand this step-by-step:
First, expand [tex]\((x + 2i)(x - 2i)\)[/tex]:
[tex]\[(x + 2i)(x - 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4\][/tex]
Next, expand [tex]\((x + 3i)(x - 3i)\)[/tex]:
[tex]\[(x + 3i)(x - 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9\][/tex]
Now multiply [tex]\((x^2 + 4)\)[/tex] by [tex]\((x^2 + 9)\)[/tex]:
[tex]\[ (x^2 + 4)(x^2 + 9) = x^4 + 9x^2 + 4x^2 + 36 \][/tex]
[tex]\[ = x^4 + 13x^2 + 36\][/tex]
The leading coefficient of the polynomial [tex]\( x^4 + 13x^2 + 36 \)[/tex] is 1.
### Conclusion
Based on these expansions, the polynomial function that has a leading coefficient of 1 and roots [tex]\(2i\)[/tex] and [tex]\(3i\)[/tex] with multiplicity 1 is:
[tex]\[ f(x) = (x + 2i)(x + 3i)(x - 2i)(x - 3i) \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{(x+2i)(x+3i)(x-2i)(x-3i)} \][/tex]
### Option 1: [tex]\(f(x) = (x - 2i)(x - 3i)\)[/tex]
Let's expand this:
[tex]\[(x - 2i)(x - 3i)\][/tex]
To expand, we apply the distributive property (also known as FOIL method):
[tex]\[ (x - 2i)(x - 3i) = x^2 - 3ix - 2ix + (2i \cdot 3i) \][/tex]
[tex]\[ = x^2 - 5ix + (-6) \][/tex]
[tex]\[ = x^2 - 5ix - 6 \][/tex]
### Option 2: [tex]\(f(x) = (x + 2i)(x + 3i)\)[/tex]
Let's expand this:
[tex]\[(x + 2i)(x + 3i)\][/tex]
Using the distributive property:
[tex]\[ (x + 2i)(x + 3i) = x^2 + 3ix + 2ix + (2i \cdot 3i) \][/tex]
[tex]\[ = x^2 + 5ix + (-6) \][/tex]
[tex]\[ = x^2 + 5ix - 6 \][/tex]
### Option 3: [tex]\(f(x) = (x - 2)(x - 3)(x - 2i)(x - 3i)\)[/tex]
Let's expand this step-by-step:
First, expand [tex]\((x - 2)(x - 3)\)[/tex]:
[tex]\[(x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6\][/tex]
Next, expand [tex]\((x - 2i)(x - 3i)\)[/tex] (we did this part in Option 1):
[tex]\[(x - 2i)(x - 3i) = x^2 - 5ix - 6\][/tex]
Now multiply [tex]\( (x^2 - 5x + 6) \)[/tex] by [tex]\((x^2 - 5ix - 6)\)[/tex]:
Without completing this entire expansion, observe that it introduces non-real coefficients and mix polynomial degrees, making it more complex than the desired result.
### Option 4: [tex]\(f(x) = (x + 2i)(x + 3i)(x - 2i)(x - 3i)\)[/tex]
Let's expand this step-by-step:
First, expand [tex]\((x + 2i)(x - 2i)\)[/tex]:
[tex]\[(x + 2i)(x - 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4\][/tex]
Next, expand [tex]\((x + 3i)(x - 3i)\)[/tex]:
[tex]\[(x + 3i)(x - 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9\][/tex]
Now multiply [tex]\((x^2 + 4)\)[/tex] by [tex]\((x^2 + 9)\)[/tex]:
[tex]\[ (x^2 + 4)(x^2 + 9) = x^4 + 9x^2 + 4x^2 + 36 \][/tex]
[tex]\[ = x^4 + 13x^2 + 36\][/tex]
The leading coefficient of the polynomial [tex]\( x^4 + 13x^2 + 36 \)[/tex] is 1.
### Conclusion
Based on these expansions, the polynomial function that has a leading coefficient of 1 and roots [tex]\(2i\)[/tex] and [tex]\(3i\)[/tex] with multiplicity 1 is:
[tex]\[ f(x) = (x + 2i)(x + 3i)(x - 2i)(x - 3i) \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{(x+2i)(x+3i)(x-2i)(x-3i)} \][/tex]
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