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Sagot :
To find the polynomial function of the lowest degree with a leading coefficient of 1 and roots [tex]\(i\)[/tex], [tex]\(-2\)[/tex], and [tex]\(2\)[/tex], let's follow these steps:
### Step 1: Understand the Roots
The roots given are [tex]\(i\)[/tex] (the imaginary unit), [tex]\(-2\)[/tex], and [tex]\(2\)[/tex]. A polynomial with these roots will be:
[tex]\[ f(x) = (x - i)(x + 2)(x - 2) \][/tex]
### Step 2: Create Factors from the Roots
Each root provides a factor of the polynomial:
1. Root [tex]\(i\)[/tex] gives the factor [tex]\((x - i)\)[/tex].
2. Root [tex]\(-2\)[/tex] gives the factor [tex]\((x + 2)\)[/tex].
3. Root [tex]\(2\)[/tex] gives the factor [tex]\((x - 2)\)[/tex].
The polynomial is given by the product of these factors:
[tex]\[ f(x) = (x - i)(x + 2)(x - 2) \][/tex]
### Step 3: Multiply the Factors
Let's start by multiplying two of the factors first:
[tex]\[ (x + 2)(x - 2) \][/tex]
Using the difference of squares:
[tex]\[ (x + 2)(x - 2) = x^2 - 4 \][/tex]
Now, multiply the result by the remaining factor:
[tex]\[ f(x) = (x - i)(x^2 - 4) \][/tex]
### Step 4: Expand the Polynomial
Next, expand the expression:
[tex]\[ f(x) = x(x^2 - 4) - i(x^2 - 4) \][/tex]
[tex]\[ f(x) = x^3 - 4x - ix^2 + 4i \][/tex]
So the expanded polynomial is:
[tex]\[ f(x) = x^3 - ix^2 - 4x + 4i \][/tex]
### Step 5: Check Coefficients and Simplify
We need to ensure the polynomial is in standard form with real coefficients where possible. The polynomial we obtained is:
[tex]\[ f(x) = x^3 - ix^2 - 4x + 4i \][/tex]
This matches the polynomial:
[tex]\[ x^3 - ix^2 - 4x + 4i \][/tex]
Now, let’s observe the choices given:
1. [tex]\(f(x)=x^3-x^2-4x+4\)[/tex]
2. [tex]\(f(x)=x^4-3x^2-4\)[/tex]
3. [tex]\(f(x)=x^4+3x^2-4\)[/tex]
4. [tex]\(f(x)=x^3+x^2-4x-4\)[/tex]
The polynomial that matches our expression [tex]\(x^3 - ix^2 - 4x + 4i\)[/tex] (considering the imaginary unit [tex]\(i\)[/tex]) is not exactly listed as one of the standard polynomial forms with real coefficients without imaginary parts. The correct polynomial form we should use, considering the inclusion of [tex]\(i\)[/tex] and standard usage, is:
[tex]\[ f(x) = x^3 - i x^2 - 4x + 4i \][/tex]
But from given options, none exactly matches, so the required polynomial function of lowest degree with lead coefficient 1 and roots [tex]\(i, -2, 2\)[/tex] is not provided exactly in real numerical part options. Therefore, considering the exact polynomial structure observed ensures":
[tex]\[ P(x) = x^3 - i x^2 - 4 + 4i \][/tex].
### Step 1: Understand the Roots
The roots given are [tex]\(i\)[/tex] (the imaginary unit), [tex]\(-2\)[/tex], and [tex]\(2\)[/tex]. A polynomial with these roots will be:
[tex]\[ f(x) = (x - i)(x + 2)(x - 2) \][/tex]
### Step 2: Create Factors from the Roots
Each root provides a factor of the polynomial:
1. Root [tex]\(i\)[/tex] gives the factor [tex]\((x - i)\)[/tex].
2. Root [tex]\(-2\)[/tex] gives the factor [tex]\((x + 2)\)[/tex].
3. Root [tex]\(2\)[/tex] gives the factor [tex]\((x - 2)\)[/tex].
The polynomial is given by the product of these factors:
[tex]\[ f(x) = (x - i)(x + 2)(x - 2) \][/tex]
### Step 3: Multiply the Factors
Let's start by multiplying two of the factors first:
[tex]\[ (x + 2)(x - 2) \][/tex]
Using the difference of squares:
[tex]\[ (x + 2)(x - 2) = x^2 - 4 \][/tex]
Now, multiply the result by the remaining factor:
[tex]\[ f(x) = (x - i)(x^2 - 4) \][/tex]
### Step 4: Expand the Polynomial
Next, expand the expression:
[tex]\[ f(x) = x(x^2 - 4) - i(x^2 - 4) \][/tex]
[tex]\[ f(x) = x^3 - 4x - ix^2 + 4i \][/tex]
So the expanded polynomial is:
[tex]\[ f(x) = x^3 - ix^2 - 4x + 4i \][/tex]
### Step 5: Check Coefficients and Simplify
We need to ensure the polynomial is in standard form with real coefficients where possible. The polynomial we obtained is:
[tex]\[ f(x) = x^3 - ix^2 - 4x + 4i \][/tex]
This matches the polynomial:
[tex]\[ x^3 - ix^2 - 4x + 4i \][/tex]
Now, let’s observe the choices given:
1. [tex]\(f(x)=x^3-x^2-4x+4\)[/tex]
2. [tex]\(f(x)=x^4-3x^2-4\)[/tex]
3. [tex]\(f(x)=x^4+3x^2-4\)[/tex]
4. [tex]\(f(x)=x^3+x^2-4x-4\)[/tex]
The polynomial that matches our expression [tex]\(x^3 - ix^2 - 4x + 4i\)[/tex] (considering the imaginary unit [tex]\(i\)[/tex]) is not exactly listed as one of the standard polynomial forms with real coefficients without imaginary parts. The correct polynomial form we should use, considering the inclusion of [tex]\(i\)[/tex] and standard usage, is:
[tex]\[ f(x) = x^3 - i x^2 - 4x + 4i \][/tex]
But from given options, none exactly matches, so the required polynomial function of lowest degree with lead coefficient 1 and roots [tex]\(i, -2, 2\)[/tex] is not provided exactly in real numerical part options. Therefore, considering the exact polynomial structure observed ensures":
[tex]\[ P(x) = x^3 - i x^2 - 4 + 4i \][/tex].
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