Answer :
Sure! Let's factor the given polynomials by grouping.
### Problem 2:
Factor the polynomial [tex]\(8h^3 + 4h^2 + 10h + 5\)[/tex].
Steps:
1. Group terms: Based on the given hint, group as [tex]\((8h^3 + 4h^2) + (10h + 5)\)[/tex].
2. Factor each group independently:
- First group: [tex]\(8h^3 + 4h^2\)[/tex]
- Factor out the greatest common factor (GCF), which is [tex]\(4h^2\)[/tex]:
- [tex]\(4h^2(2h + 1)\)[/tex]
- Second group: [tex]\(10h + 5\)[/tex]
- Factor out the GCF, which is [tex]\(5\)[/tex]:
- [tex]\(5(2h + 1)\)[/tex]
3. Combine the factored terms:
Both groups have a common factor [tex]\((2h + 1)\)[/tex], so we combine them:
- [tex]\((2h + 1)(4h^2 + 5)\)[/tex]
### Problem 3:
Factor the polynomial [tex]\(14x^5 + 35x^3 - 4x^2 - 10\)[/tex].
Steps:
1. Group terms: [tex]\((14x^5 + 35x^3) + (-4x^2 - 10)\)[/tex].
2. Factor each group independently:
- First group: [tex]\(14x^5 + 35x^3\)[/tex]
- Factor out [tex]\(7x^3\)[/tex]:
- [tex]\(7x^3(2x^2 + 5)\)[/tex]
- Second group: [tex]\(-4x^2 - 10\)[/tex]
- Factor out [tex]\(-2\)[/tex]:
- [tex]\(-2(2x^2 + 5)\)[/tex]
3. Combine the factored terms:
Both groups have a common factor [tex]\((2x^2 + 5)\)[/tex], so we can combine:
- [tex]\((2x^2 + 5)(7x^3 - 2)\)[/tex]
### Problem 4:
Factor the polynomial [tex]\(15a^2b - 25a^2 + 9b - 15\)[/tex].
Steps:
1. Group terms: [tex]\((15a^2b - 25a^2) + (9b - 15)\)[/tex].
2. Factor each group independently:
- First group: [tex]\(15a^2b - 25a^2\)[/tex]
- Factor out [tex]\(5a^2\)[/tex]:
- [tex]\(5a^2(3b - 5)\)[/tex]
- Second group: [tex]\(9b - 15\)[/tex]
- Factor out [tex]\(3\)[/tex]:
- [tex]\(3(3b - 5)\)[/tex]
3. Combine the factored terms:
Both groups have a common factor [tex]\((3b - 5)\)[/tex], so we can combine:
- [tex]\((3b - 5)(5a^2 + 3)\)[/tex]
That's how each polynomial is factored by grouping!
### Problem 2:
Factor the polynomial [tex]\(8h^3 + 4h^2 + 10h + 5\)[/tex].
Steps:
1. Group terms: Based on the given hint, group as [tex]\((8h^3 + 4h^2) + (10h + 5)\)[/tex].
2. Factor each group independently:
- First group: [tex]\(8h^3 + 4h^2\)[/tex]
- Factor out the greatest common factor (GCF), which is [tex]\(4h^2\)[/tex]:
- [tex]\(4h^2(2h + 1)\)[/tex]
- Second group: [tex]\(10h + 5\)[/tex]
- Factor out the GCF, which is [tex]\(5\)[/tex]:
- [tex]\(5(2h + 1)\)[/tex]
3. Combine the factored terms:
Both groups have a common factor [tex]\((2h + 1)\)[/tex], so we combine them:
- [tex]\((2h + 1)(4h^2 + 5)\)[/tex]
### Problem 3:
Factor the polynomial [tex]\(14x^5 + 35x^3 - 4x^2 - 10\)[/tex].
Steps:
1. Group terms: [tex]\((14x^5 + 35x^3) + (-4x^2 - 10)\)[/tex].
2. Factor each group independently:
- First group: [tex]\(14x^5 + 35x^3\)[/tex]
- Factor out [tex]\(7x^3\)[/tex]:
- [tex]\(7x^3(2x^2 + 5)\)[/tex]
- Second group: [tex]\(-4x^2 - 10\)[/tex]
- Factor out [tex]\(-2\)[/tex]:
- [tex]\(-2(2x^2 + 5)\)[/tex]
3. Combine the factored terms:
Both groups have a common factor [tex]\((2x^2 + 5)\)[/tex], so we can combine:
- [tex]\((2x^2 + 5)(7x^3 - 2)\)[/tex]
### Problem 4:
Factor the polynomial [tex]\(15a^2b - 25a^2 + 9b - 15\)[/tex].
Steps:
1. Group terms: [tex]\((15a^2b - 25a^2) + (9b - 15)\)[/tex].
2. Factor each group independently:
- First group: [tex]\(15a^2b - 25a^2\)[/tex]
- Factor out [tex]\(5a^2\)[/tex]:
- [tex]\(5a^2(3b - 5)\)[/tex]
- Second group: [tex]\(9b - 15\)[/tex]
- Factor out [tex]\(3\)[/tex]:
- [tex]\(3(3b - 5)\)[/tex]
3. Combine the factored terms:
Both groups have a common factor [tex]\((3b - 5)\)[/tex], so we can combine:
- [tex]\((3b - 5)(5a^2 + 3)\)[/tex]
That's how each polynomial is factored by grouping!
