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Sure! Let's go through the process of factoring each polynomial by grouping.
41. [tex]\(x^3 + x^2 + x + 1\)[/tex]:
1. Group terms: [tex]\((x^3 + x^2) + (x + 1)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(x + 1)\)[/tex].
- The second group is already [tex]\((x + 1)\)[/tex].
3. Combine common factors: Notice [tex]\((x + 1)\)[/tex] is a common factor:
[tex]\((x + 1)(x^2 + 1)\)[/tex].
42. [tex]\(10x^3 + 20x^2 + x + 2\)[/tex]:
1. Group terms: [tex]\((10x^3 + 20x^2) + (x + 2)\)[/tex].
2. Factor each group:
- From the first group, factor out a [tex]\(10x^2\)[/tex]: [tex]\(10x^2(x + 2)\)[/tex].
- The second group does not have a common factor, so it remains [tex]\((x + 2)\)[/tex].
3. Combine common factors: Factoring out the common [tex]\((x + 2)\)[/tex]:
[tex]\((x + 2)(10x^2 + 1)\)[/tex].
43. [tex]\(x^3 + 3x^2 + 10x + 30\)[/tex]:
1. Group terms: [tex]\((x^3 + 3x^2) + (10x + 30)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(x + 3)\)[/tex].
- From the second group, factor out a [tex]\(10\)[/tex]: [tex]\(10(x + 3)\)[/tex].
3. Combine common factors: Notice the common [tex]\((x + 3)\)[/tex]:
[tex]\((x + 3)(x^2 + 10)\)[/tex].
44. [tex]\(x^3 - 2x^2 + 4x - 8\)[/tex]:
1. Group terms: [tex]\((x^3 - 2x^2) + (4x - 8)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(x - 2)\)[/tex].
- From the second group, factor out a [tex]\(4\)[/tex]: [tex]\(4(x - 2)\)[/tex].
3. Combine common factors: The common factor is [tex]\((x - 2)\)[/tex]:
[tex]\((x - 2)(x^2 + 4)\)[/tex].
45. [tex]\(2x^3 - 5x^2 + 18x - 45\)[/tex]:
1. Group terms: [tex]\((2x^3 - 5x^2) + (18x - 45)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(2x - 5)\)[/tex].
- From the second group, factor out a [tex]\(9\)[/tex]: [tex]\(9(2x - 5)\)[/tex].
3. Combine common factors: Again, [tex]\((2x - 5)\)[/tex] is common:
[tex]\((2x - 5)(x^2 + 9)\)[/tex].
46. [tex]\(-2x^3 - 4x^2 - 3x - 6\)[/tex]:
1. Group terms: [tex]\((-2x^3 - 4x^2) + (-3x - 6)\)[/tex].
2. Factor each group:
- From the first group, factor out [tex]\(-2x^2\)[/tex]: [tex]\(-2x^2(x + 2)\)[/tex].
- From the second group, factor out [tex]\(-3\)[/tex]: [tex]\(-3(x + 2)\)[/tex].
3. Combine common factors: Notice the common [tex]\((x + 2)\)[/tex]:
[tex]\(-(x + 2)(2x^2 + 3)\)[/tex].
47. [tex]\(3x^3 - 6x^2 + x - 2\)[/tex]:
1. Group terms: [tex]\((3x^3 - 6x^2) + (x - 2)\)[/tex].
2. Factor each group:
- From the first group, factor out a [tex]\(3x^2\)[/tex]: [tex]\(3x^2(x - 2)\)[/tex].
- The second group does not factor more than [tex]\(x - 2\)[/tex].
3. Combine common factors: Therefore, [tex]\((x - 2)(3x^2 + 1)\)[/tex].
48. [tex]\(2x^3 - x^2 + 2x - 1\)[/tex]:
1. Group terms: [tex]\((2x^3 - x^2) + (2x - 1)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(2x - 1)\)[/tex].
- The second group remains [tex]\((2x - 1)\)[/tex].
3. Combine common factors: Here, [tex]\((2x - 1)\)[/tex] is the common factor:
[tex]\((2x - 1)(x^2 + 1)\)[/tex].
49. [tex]\(3x^3 - 2x^2 - 9x + 6\)[/tex]:
1. Group terms: [tex]\((3x^3 - 2x^2) + (-9x + 6)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(3x - 2)\)[/tex].
- From the second group, factor out a [tex]\(-3\)[/tex]: [tex]\(-3(3x - 2)\)[/tex].
3. Combine common factors: The common factor is [tex]\((3x - 2)\)[/tex]:
[tex]\((3x - 2)(x^2 - 3)\)[/tex].
These steps allow us to group and factor the polynomials efficiently by identifying common factors within grouped terms.
41. [tex]\(x^3 + x^2 + x + 1\)[/tex]:
1. Group terms: [tex]\((x^3 + x^2) + (x + 1)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(x + 1)\)[/tex].
- The second group is already [tex]\((x + 1)\)[/tex].
3. Combine common factors: Notice [tex]\((x + 1)\)[/tex] is a common factor:
[tex]\((x + 1)(x^2 + 1)\)[/tex].
42. [tex]\(10x^3 + 20x^2 + x + 2\)[/tex]:
1. Group terms: [tex]\((10x^3 + 20x^2) + (x + 2)\)[/tex].
2. Factor each group:
- From the first group, factor out a [tex]\(10x^2\)[/tex]: [tex]\(10x^2(x + 2)\)[/tex].
- The second group does not have a common factor, so it remains [tex]\((x + 2)\)[/tex].
3. Combine common factors: Factoring out the common [tex]\((x + 2)\)[/tex]:
[tex]\((x + 2)(10x^2 + 1)\)[/tex].
43. [tex]\(x^3 + 3x^2 + 10x + 30\)[/tex]:
1. Group terms: [tex]\((x^3 + 3x^2) + (10x + 30)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(x + 3)\)[/tex].
- From the second group, factor out a [tex]\(10\)[/tex]: [tex]\(10(x + 3)\)[/tex].
3. Combine common factors: Notice the common [tex]\((x + 3)\)[/tex]:
[tex]\((x + 3)(x^2 + 10)\)[/tex].
44. [tex]\(x^3 - 2x^2 + 4x - 8\)[/tex]:
1. Group terms: [tex]\((x^3 - 2x^2) + (4x - 8)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(x - 2)\)[/tex].
- From the second group, factor out a [tex]\(4\)[/tex]: [tex]\(4(x - 2)\)[/tex].
3. Combine common factors: The common factor is [tex]\((x - 2)\)[/tex]:
[tex]\((x - 2)(x^2 + 4)\)[/tex].
45. [tex]\(2x^3 - 5x^2 + 18x - 45\)[/tex]:
1. Group terms: [tex]\((2x^3 - 5x^2) + (18x - 45)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(2x - 5)\)[/tex].
- From the second group, factor out a [tex]\(9\)[/tex]: [tex]\(9(2x - 5)\)[/tex].
3. Combine common factors: Again, [tex]\((2x - 5)\)[/tex] is common:
[tex]\((2x - 5)(x^2 + 9)\)[/tex].
46. [tex]\(-2x^3 - 4x^2 - 3x - 6\)[/tex]:
1. Group terms: [tex]\((-2x^3 - 4x^2) + (-3x - 6)\)[/tex].
2. Factor each group:
- From the first group, factor out [tex]\(-2x^2\)[/tex]: [tex]\(-2x^2(x + 2)\)[/tex].
- From the second group, factor out [tex]\(-3\)[/tex]: [tex]\(-3(x + 2)\)[/tex].
3. Combine common factors: Notice the common [tex]\((x + 2)\)[/tex]:
[tex]\(-(x + 2)(2x^2 + 3)\)[/tex].
47. [tex]\(3x^3 - 6x^2 + x - 2\)[/tex]:
1. Group terms: [tex]\((3x^3 - 6x^2) + (x - 2)\)[/tex].
2. Factor each group:
- From the first group, factor out a [tex]\(3x^2\)[/tex]: [tex]\(3x^2(x - 2)\)[/tex].
- The second group does not factor more than [tex]\(x - 2\)[/tex].
3. Combine common factors: Therefore, [tex]\((x - 2)(3x^2 + 1)\)[/tex].
48. [tex]\(2x^3 - x^2 + 2x - 1\)[/tex]:
1. Group terms: [tex]\((2x^3 - x^2) + (2x - 1)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(2x - 1)\)[/tex].
- The second group remains [tex]\((2x - 1)\)[/tex].
3. Combine common factors: Here, [tex]\((2x - 1)\)[/tex] is the common factor:
[tex]\((2x - 1)(x^2 + 1)\)[/tex].
49. [tex]\(3x^3 - 2x^2 - 9x + 6\)[/tex]:
1. Group terms: [tex]\((3x^3 - 2x^2) + (-9x + 6)\)[/tex].
2. Factor each group:
- From the first group, factor out an [tex]\(x^2\)[/tex]: [tex]\(x^2(3x - 2)\)[/tex].
- From the second group, factor out a [tex]\(-3\)[/tex]: [tex]\(-3(3x - 2)\)[/tex].
3. Combine common factors: The common factor is [tex]\((3x - 2)\)[/tex]:
[tex]\((3x - 2)(x^2 - 3)\)[/tex].
These steps allow us to group and factor the polynomials efficiently by identifying common factors within grouped terms.
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