Sure, let's simplify each expression step by step.
### 7. [tex]\(8a^3 - 6a^2\)[/tex]
To factor out the common term:
- Common factor: [tex]\(a^2\)[/tex].
[tex]\[
8a^3 - 6a^2 = a^2 (8a - 6)
\][/tex]
### 9. [tex]\(b^4 - b^3\)[/tex]
To factor out the common term:
- Common factor: [tex]\(b^3\)[/tex].
[tex]\[
b^4 - b^3 = b^3 (b - 1)
\][/tex]
### 11. [tex]\(14a - 21b + 35\)[/tex]
To factor out the common term:
- Common factor: 7.
[tex]\[
14a - 21b + 35 = 7 (2a - 3b + 5)
\][/tex]
### 13. [tex]\(20x - 12xy + 4xz\)[/tex]
To factor out the common term:
- Common factor: [tex]\(4x\)[/tex].
[tex]\[
20x - 12xy + 4xz = 4x (5 - 3y + z)
\][/tex]
### 15. [tex]\(10x^2y - 15xy^2 + 25xy\)[/tex]
To factor out the common term:
- Common factor: [tex]\(5xy\)[/tex].
[tex]\[
10x^2y - 15xy^2 + 25xy = 5xy (2x - 3y + 5)
\][/tex]
### 17. [tex]\(2x^2 + 6x + 8x^3 - 12x^4\)[/tex]
To factor out the common term:
- Common factor: [tex]\(2x\)[/tex].
[tex]\[
2x^2 + 6x + 8x^3 - 12x^4 = 2x (x + 3 + 4x^2 - 6x^3)
\][/tex]
### 19. [tex]\(m^3 n^2 p^4 + m^4 n^3 p^5 - m^6 n^4 p^4 + m^2 n^4 p^3\)[/tex]
There isn't a single common factor for all terms. However, let's write them in a standard form:
- Grouping like terms:
[tex]\[
m^3 n^2 p^4 + m^4 n^3 p^5 - m^6 n^4 p^4 + m^2 n^4 p^3
\][/tex]
These expressions cannot be factored further easily, so we'll leave them in their current form:
[tex]\[
-m^6 n^4 p^4 + m^4 n^3 p^5 + m^3 n^2 p^4 + m^2 n^4 p^3
\][/tex]
To summarize, the simplified step-by-step solutions for each expression are:
1. [tex]\(8a^3 - 6a^2 = a^2 (8a - 6)\)[/tex]
2. [tex]\(b^4 - b^3 = b^3 (b - 1)\)[/tex]
3. [tex]\(14a - 21b + 35 = 7 (2a - 3b + 5)\)[/tex]
4. [tex]\(20x - 12xy + 4xz = 4x (5 - 3y + z)\)[/tex]
5. [tex]\(10x^2y - 15xy^2 + 25xy = 5xy (2x - 3y + 5)\)[/tex]
6. [tex]\(2x^2 + 6x + 8x^3 - 12x^4 = 2x (x + 3 + 4x^2 - 6x^3)\)[/tex]
7. [tex]\(m^3 n^2 p^4 + m^4 n^3 p^5 - m^6 n^4 p^4 + m^2 n^4 p^3 = -m^6 n^4 p^4 + m^4 n^3 p^5 + m^3 n^2 p^4 + m^2 n^4 p^3\)[/tex]
