Answered

Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.

What is the polynomial function of the lowest degree with leading coefficient 1 and roots [tex]i[/tex], [tex]-2[/tex], and [tex]2[/tex]?

A. [tex]f(x) = x^3 - x^2 - 4x + 4[/tex]
B. [tex]f(x) = x^4 - 3x^2 - 4[/tex]
C. [tex]f(x) = x^4 + 3x^2 - 4[/tex]
D. [tex]f(x) = x^3 + x^2 - 4x - 4[/tex]

Sagot :

GinnyAnswer
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].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.