Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To find the polynomial function of lowest degree with lead coefficient 1 and roots [tex]\(i, -2\)[/tex], and [tex]\(2\)[/tex], we need to follow these steps:
1. Identify the given roots: The roots of the polynomial are [tex]\(i, -2\)[/tex], and [tex]\(2\)[/tex].
2. Include the complex conjugate: Since [tex]\(i\)[/tex] is a root of the polynomial, its complex conjugate [tex]\(-i\)[/tex] must also be a root. Therefore, the complete set of roots is [tex]\(i, -i, -2,\)[/tex] and [tex]\(2\)[/tex].
3. Form the factors: Each root gives a factor of the polynomial:
- For [tex]\(i\)[/tex], the factor is [tex]\((x - i)\)[/tex].
- For [tex]\(-i\)[/tex], the factor is [tex]\((x + i)\)[/tex].
- For [tex]\(-2\)[/tex], the factor is [tex]\((x + 2)\)[/tex].
- For [tex]\(2\)[/tex], the factor is [tex]\((x - 2)\)[/tex].
4. Multiply the factors in pairs: Expand the polynomial step-by-step by multiplying these factors.
- First, multiply the factors involving [tex]\(i\)[/tex]:
[tex]\[ (x - i)(x + i) = x^2 - i^2 \][/tex]
Note that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[ (x - i)(x + i) = x^2 - (-1) = x^2 + 1 \][/tex]
- Next, multiply the factors involving [tex]\(-2\)[/tex] and [tex]\(2\)[/tex]:
[tex]\[ (x + 2)(x - 2) = x^2 - 4 \][/tex]
5. Combine the results: Now, multiply the two quadratic expressions obtained:
[tex]\[ (x^2 + 1)(x^2 - 4) \][/tex]
6. Expand the product:
[tex]\[ (x^2 + 1)(x^2 - 4) = x^2(x^2 - 4) + 1(x^2 - 4) \][/tex]
Simplify the expression:
[tex]\[ x^2(x^2 - 4) + 1(x^2 - 4) = x^4 - 4x^2 + x^2 - 4 = x^4 - 3x^2 - 4 \][/tex]
Thus, the polynomial of the lowest degree with lead coefficient 1 and roots [tex]\(i, -i, -2\)[/tex], and [tex]\(2\)[/tex] is:
[tex]\[ f(x) = x^4 - 3x^2 - 4 \][/tex]
So the correct answer is:
[tex]\[ \boxed{f(x) = x^4 - 3 x^2 - 4} \][/tex]
1. Identify the given roots: The roots of the polynomial are [tex]\(i, -2\)[/tex], and [tex]\(2\)[/tex].
2. Include the complex conjugate: Since [tex]\(i\)[/tex] is a root of the polynomial, its complex conjugate [tex]\(-i\)[/tex] must also be a root. Therefore, the complete set of roots is [tex]\(i, -i, -2,\)[/tex] and [tex]\(2\)[/tex].
3. Form the factors: Each root gives a factor of the polynomial:
- For [tex]\(i\)[/tex], the factor is [tex]\((x - i)\)[/tex].
- For [tex]\(-i\)[/tex], the factor is [tex]\((x + i)\)[/tex].
- For [tex]\(-2\)[/tex], the factor is [tex]\((x + 2)\)[/tex].
- For [tex]\(2\)[/tex], the factor is [tex]\((x - 2)\)[/tex].
4. Multiply the factors in pairs: Expand the polynomial step-by-step by multiplying these factors.
- First, multiply the factors involving [tex]\(i\)[/tex]:
[tex]\[ (x - i)(x + i) = x^2 - i^2 \][/tex]
Note that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[ (x - i)(x + i) = x^2 - (-1) = x^2 + 1 \][/tex]
- Next, multiply the factors involving [tex]\(-2\)[/tex] and [tex]\(2\)[/tex]:
[tex]\[ (x + 2)(x - 2) = x^2 - 4 \][/tex]
5. Combine the results: Now, multiply the two quadratic expressions obtained:
[tex]\[ (x^2 + 1)(x^2 - 4) \][/tex]
6. Expand the product:
[tex]\[ (x^2 + 1)(x^2 - 4) = x^2(x^2 - 4) + 1(x^2 - 4) \][/tex]
Simplify the expression:
[tex]\[ x^2(x^2 - 4) + 1(x^2 - 4) = x^4 - 4x^2 + x^2 - 4 = x^4 - 3x^2 - 4 \][/tex]
Thus, the polynomial of the lowest degree with lead coefficient 1 and roots [tex]\(i, -i, -2\)[/tex], and [tex]\(2\)[/tex] is:
[tex]\[ f(x) = x^4 - 3x^2 - 4 \][/tex]
So the correct answer is:
[tex]\[ \boxed{f(x) = x^4 - 3 x^2 - 4} \][/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
