At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To write a polynomial function of least degree with real coefficients in standard form that has the given zeros: [tex]\(-2\)[/tex], [tex]\(-4\)[/tex], and [tex]\(-3+4i\)[/tex], follow these step-by-step instructions:
1. Identify the zeros and their conjugates:
- Zeros: [tex]\(-2\)[/tex], [tex]\(-4\)[/tex], and [tex]\(-3+4i\)[/tex].
- Since the coefficients must be real, the conjugate of the complex zero [tex]\(-3+4i\)[/tex] must also be included. Thus, the conjugate is [tex]\(-3-4i\)[/tex].
2. Express each zero as a factor:
- For each zero [tex]\(a\)[/tex], the corresponding factor is [tex]\((x - a)\)[/tex].
- Therefore, the factors are:
[tex]\[ (x + 2), (x + 4), (x + 3 - 4i), \text{ and } (x + 3 + 4i) \][/tex]
3. Combine the complex conjugate factors into a quadratic factor:
- Multiply the conjugate pair:
[tex]\[ (x + 3 - 4i)(x + 3 + 4i) \][/tex]
- This can be simplified using the difference of squares:
[tex]\[ (x + 3)^2 - (4i)^2 \][/tex]
- Calculating inside:
[tex]\[ (x + 3)^2 = x^2 + 6x + 9 \][/tex]
[tex]\[ (4i)^2 = 16i^2 = -16 \][/tex]
- Putting it together:
[tex]\[ (x + 3)^2 - (4i)^2 = x^2 + 6x + 9 - (-16) = x^2 + 6x + 25 \][/tex]
4. Multiply all the factors together:
- Now multiply the remaining linear factors by this quadratic factor:
[tex]\[ (x + 2)(x + 4)(x^2 + 6x + 25) \][/tex]
5. First, multiply the linear factors:
- Compute [tex]\((x + 2)(x + 4)\)[/tex]:
[tex]\[ (x + 2)(x + 4) = x^2 + 4x + 2x + 8 = x^2 + 6x + 8 \][/tex]
6. Next, multiply this result by the quadratic factor:
- Compute [tex]\((x^2 + 6x + 8)(x^2 + 6x + 25)\)[/tex]:
[tex]\[ (x^2 + 6x + 8)(x^2 + 6x + 25) \][/tex]
7. Perform the polynomial multiplication:
- Multiply each term in [tex]\((x^2 + 6x + 8)\)[/tex] by each term in [tex]\((x^2 + 6x + 25)\)[/tex]:
- [tex]\(x^2 \cdot x^2 = x^4\)[/tex]
- [tex]\(x^2 \cdot 6x = 6x^3\)[/tex]
- [tex]\(x^2 \cdot 25 = 25x^2\)[/tex]
- [tex]\(6x \cdot x^2 = 6x^3\)[/tex]
- [tex]\(6x \cdot 6x = 36x^2\)[/tex]
- [tex]\(6x \cdot 25 = 150x\)[/tex]
- [tex]\(8 \cdot x^2 = 8x^2\)[/tex]
- [tex]\(8 \cdot 6x = 48x\)[/tex]
- [tex]\(8 \cdot 25 = 200\)[/tex]
- Summing all these terms:
[tex]\[ x^4 + 12x^3 + (25x^2 + 36x^2 + 8x^2) + (150x + 48x) + 200 \][/tex]
- Combine like terms:
[tex]\[ x^4 + 12x^3 + 69x^2 + 198x + 200 \][/tex]
Therefore, the polynomial function of least degree with real coefficients in standard form that has the given zeros is:
[tex]\[ \boxed{x^4 + 12x^3 + 69x^2 + 198x + 200} \][/tex]
1. Identify the zeros and their conjugates:
- Zeros: [tex]\(-2\)[/tex], [tex]\(-4\)[/tex], and [tex]\(-3+4i\)[/tex].
- Since the coefficients must be real, the conjugate of the complex zero [tex]\(-3+4i\)[/tex] must also be included. Thus, the conjugate is [tex]\(-3-4i\)[/tex].
2. Express each zero as a factor:
- For each zero [tex]\(a\)[/tex], the corresponding factor is [tex]\((x - a)\)[/tex].
- Therefore, the factors are:
[tex]\[ (x + 2), (x + 4), (x + 3 - 4i), \text{ and } (x + 3 + 4i) \][/tex]
3. Combine the complex conjugate factors into a quadratic factor:
- Multiply the conjugate pair:
[tex]\[ (x + 3 - 4i)(x + 3 + 4i) \][/tex]
- This can be simplified using the difference of squares:
[tex]\[ (x + 3)^2 - (4i)^2 \][/tex]
- Calculating inside:
[tex]\[ (x + 3)^2 = x^2 + 6x + 9 \][/tex]
[tex]\[ (4i)^2 = 16i^2 = -16 \][/tex]
- Putting it together:
[tex]\[ (x + 3)^2 - (4i)^2 = x^2 + 6x + 9 - (-16) = x^2 + 6x + 25 \][/tex]
4. Multiply all the factors together:
- Now multiply the remaining linear factors by this quadratic factor:
[tex]\[ (x + 2)(x + 4)(x^2 + 6x + 25) \][/tex]
5. First, multiply the linear factors:
- Compute [tex]\((x + 2)(x + 4)\)[/tex]:
[tex]\[ (x + 2)(x + 4) = x^2 + 4x + 2x + 8 = x^2 + 6x + 8 \][/tex]
6. Next, multiply this result by the quadratic factor:
- Compute [tex]\((x^2 + 6x + 8)(x^2 + 6x + 25)\)[/tex]:
[tex]\[ (x^2 + 6x + 8)(x^2 + 6x + 25) \][/tex]
7. Perform the polynomial multiplication:
- Multiply each term in [tex]\((x^2 + 6x + 8)\)[/tex] by each term in [tex]\((x^2 + 6x + 25)\)[/tex]:
- [tex]\(x^2 \cdot x^2 = x^4\)[/tex]
- [tex]\(x^2 \cdot 6x = 6x^3\)[/tex]
- [tex]\(x^2 \cdot 25 = 25x^2\)[/tex]
- [tex]\(6x \cdot x^2 = 6x^3\)[/tex]
- [tex]\(6x \cdot 6x = 36x^2\)[/tex]
- [tex]\(6x \cdot 25 = 150x\)[/tex]
- [tex]\(8 \cdot x^2 = 8x^2\)[/tex]
- [tex]\(8 \cdot 6x = 48x\)[/tex]
- [tex]\(8 \cdot 25 = 200\)[/tex]
- Summing all these terms:
[tex]\[ x^4 + 12x^3 + (25x^2 + 36x^2 + 8x^2) + (150x + 48x) + 200 \][/tex]
- Combine like terms:
[tex]\[ x^4 + 12x^3 + 69x^2 + 198x + 200 \][/tex]
Therefore, the polynomial function of least degree with real coefficients in standard form that has the given zeros is:
[tex]\[ \boxed{x^4 + 12x^3 + 69x^2 + 198x + 200} \][/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.