At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Alright, let's go through the factorization of the given polynomial expressions step-by-step to understand their factors and see how they are decomposed.
### Expression 1: [tex]\((x+1)(2+i x)(2-i x)\)[/tex]
1. Identify the roots: The factors are [tex]\((x+1)\)[/tex], [tex]\((2+ix)\)[/tex], and [tex]\((2-ix)\)[/tex].
2. The term [tex]\((x+1)\)[/tex] suggests a root at [tex]\(x = -1\)[/tex].
3. The terms [tex]\((2 + ix)\)[/tex] and [tex]\((2 - ix)\)[/tex] suggest complex roots. We can write these roots as [tex]\(x = -\frac{2}{i}\)[/tex] and [tex]\(x = \frac{2}{i}\)[/tex]. Simplifying, these become [tex]\(x = -2i\)[/tex] and [tex]\(x = 2i\)[/tex].
### Expression 2: [tex]\((x-1)(2+i x)(2-i x)\)[/tex]
1. Identify the roots: The factors are [tex]\((x-1)\)[/tex], [tex]\((2+ix)\)[/tex], and [tex]\((2-ix)\)[/tex].
2. The term [tex]\((x-1)\)[/tex] suggests a root at [tex]\(x = 1\)[/tex].
3. The terms [tex]\((2 + ix)\)[/tex] and [tex]\((2 - ix)\)[/tex] suggest complex roots [tex]\(x = -\frac{2}{i} = -2i\)[/tex] and [tex]\(x = \frac{2}{i} = 2i\)[/tex].
### Expression 3: [tex]\((x-1)[x-(3+i)][x-(3-i)]\)[/tex]
1. Identify the roots: The factors are [tex]\((x-1)\)[/tex], [tex]\((x - (3+i))\)[/tex], and [tex]\((x - (3-i))\)[/tex].
2. The term [tex]\((x-1)\)[/tex] suggests a root at [tex]\(x = 1\)[/tex].
3. The term [tex]\((x - (3+i))\)[/tex] suggests a root at [tex]\(x = 3+i\)[/tex].
4. The term [tex]\((x - (3-i))\)[/tex] suggests a root at [tex]\(x = 3-i\)[/tex].
### Expression 4: [tex]\((x+1)[x+(2+i)][x-(2-i)]\)[/tex]
1. Identify the roots: The factors are [tex]\((x+1)\)[/tex], [tex]\((x+(2+i))\)[/tex], and [tex]\((x-(2-i))\)[/tex].
2. The term [tex]\((x+1)\)[/tex] suggests a root at [tex]\(x = -1\)[/tex].
3. The term [tex]\((x+(2+i))\)[/tex] can be rewritten as [tex]\((x - (-2-i))\)[/tex], suggesting a root at [tex]\(x = -2 - i\)[/tex].
4. The term [tex]\((x-(2-i))\)[/tex] suggests a root at [tex]\(x = 2-i\)[/tex].
### Expression 5: [tex]\((x-1)[x+(2+i)][x-(2-i)]\)[/tex]
1. Identify the roots: The factors are [tex]\((x-1)\)[/tex], [tex]\((x+(2+i))\)[/tex], and [tex]\((x-(2-i))\)[/tex].
2. The term [tex]\((x-1)\)[/tex] suggests a root at [tex]\(x = 1\)[/tex].
3. The term [tex]\((x+(2+i))\)[/tex], rewritten as [tex]\((x - (-2-i))\)[/tex], suggests a root at [tex]\(x = -2 - i\)[/tex].
4. The term [tex]\((x-(2-i))\)[/tex] suggests a root at [tex]\(x = 2-i\)[/tex].
### Conclusion:
The factors provided each reflect a polynomial factored into linear terms, some of which include complex numbers. The complex conjugate pairs [tex]\((2+ix)\)[/tex] and [tex]\((2-ix)\)[/tex] or [tex]\((3+i)\)[/tex] and [tex]\((3-i)\)[/tex] suggest that these polynomials originally had complex roots. Each of these factorizations demonstrates the decomposition of a polynomial into irreducible factors.
### Expression 1: [tex]\((x+1)(2+i x)(2-i x)\)[/tex]
1. Identify the roots: The factors are [tex]\((x+1)\)[/tex], [tex]\((2+ix)\)[/tex], and [tex]\((2-ix)\)[/tex].
2. The term [tex]\((x+1)\)[/tex] suggests a root at [tex]\(x = -1\)[/tex].
3. The terms [tex]\((2 + ix)\)[/tex] and [tex]\((2 - ix)\)[/tex] suggest complex roots. We can write these roots as [tex]\(x = -\frac{2}{i}\)[/tex] and [tex]\(x = \frac{2}{i}\)[/tex]. Simplifying, these become [tex]\(x = -2i\)[/tex] and [tex]\(x = 2i\)[/tex].
### Expression 2: [tex]\((x-1)(2+i x)(2-i x)\)[/tex]
1. Identify the roots: The factors are [tex]\((x-1)\)[/tex], [tex]\((2+ix)\)[/tex], and [tex]\((2-ix)\)[/tex].
2. The term [tex]\((x-1)\)[/tex] suggests a root at [tex]\(x = 1\)[/tex].
3. The terms [tex]\((2 + ix)\)[/tex] and [tex]\((2 - ix)\)[/tex] suggest complex roots [tex]\(x = -\frac{2}{i} = -2i\)[/tex] and [tex]\(x = \frac{2}{i} = 2i\)[/tex].
### Expression 3: [tex]\((x-1)[x-(3+i)][x-(3-i)]\)[/tex]
1. Identify the roots: The factors are [tex]\((x-1)\)[/tex], [tex]\((x - (3+i))\)[/tex], and [tex]\((x - (3-i))\)[/tex].
2. The term [tex]\((x-1)\)[/tex] suggests a root at [tex]\(x = 1\)[/tex].
3. The term [tex]\((x - (3+i))\)[/tex] suggests a root at [tex]\(x = 3+i\)[/tex].
4. The term [tex]\((x - (3-i))\)[/tex] suggests a root at [tex]\(x = 3-i\)[/tex].
### Expression 4: [tex]\((x+1)[x+(2+i)][x-(2-i)]\)[/tex]
1. Identify the roots: The factors are [tex]\((x+1)\)[/tex], [tex]\((x+(2+i))\)[/tex], and [tex]\((x-(2-i))\)[/tex].
2. The term [tex]\((x+1)\)[/tex] suggests a root at [tex]\(x = -1\)[/tex].
3. The term [tex]\((x+(2+i))\)[/tex] can be rewritten as [tex]\((x - (-2-i))\)[/tex], suggesting a root at [tex]\(x = -2 - i\)[/tex].
4. The term [tex]\((x-(2-i))\)[/tex] suggests a root at [tex]\(x = 2-i\)[/tex].
### Expression 5: [tex]\((x-1)[x+(2+i)][x-(2-i)]\)[/tex]
1. Identify the roots: The factors are [tex]\((x-1)\)[/tex], [tex]\((x+(2+i))\)[/tex], and [tex]\((x-(2-i))\)[/tex].
2. The term [tex]\((x-1)\)[/tex] suggests a root at [tex]\(x = 1\)[/tex].
3. The term [tex]\((x+(2+i))\)[/tex], rewritten as [tex]\((x - (-2-i))\)[/tex], suggests a root at [tex]\(x = -2 - i\)[/tex].
4. The term [tex]\((x-(2-i))\)[/tex] suggests a root at [tex]\(x = 2-i\)[/tex].
### Conclusion:
The factors provided each reflect a polynomial factored into linear terms, some of which include complex numbers. The complex conjugate pairs [tex]\((2+ix)\)[/tex] and [tex]\((2-ix)\)[/tex] or [tex]\((3+i)\)[/tex] and [tex]\((3-i)\)[/tex] suggest that these polynomials originally had complex roots. Each of these factorizations demonstrates the decomposition of a polynomial into irreducible factors.
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. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.