Solved] Please Explain And Answer The Questions And Thank, 47% OFF - Free Printable

To solve the problem, we need to factor each polynomial in Column A by taking out the greatest common monomial factor. Let's go through each polynomial step by step.

Column A:

#### 1. \( 4x + 6 \)
- The greatest common factor (GCF) of \( 4x \) and \( 6 \) is \( 2 \).
- Factor out \( 2 \):
\[
4x + 6 = 2(2x + 3)
\]
- Match: M. \( 2(2x + 3) \)

#### 2. \( 2x^2 - 5x \)
- The GCF of \( 2x^2 \) and \( -5x \) is \( x \).
- Factor out \( x \):
\[
2x^2 - 5x = x(2x - 5)
\]
- Match: K. \( x(2x - 5) \)

#### 3. \( 14x^5 + 21x^3 \)
- The GCF of \( 14x^5 \) and \( 21x^3 \) is \( 7x^3 \).
- Factor out \( 7x^3 \):
\[
14x^5 + 21x^3 = 7x^3(2x^2 + 3)
\]
- Match: H. \( 7x^3(2x^2 + 3) \)

#### 4. \( x^2y + x^6y^2 - x^4y^4 \)
- The GCF of \( x^2y \), \( x^6y^2 \), and \( -x^4y^4 \) is \( x^2y \).
- Factor out \( x^2y \):
\[
x^2y + x^6y^2 - x^4y^4 = x^2y(1 + x^4y - x^2y^3)
\]
- Match: F. \( x^2y(1 + x^4y - x^2y^3) \)

#### 5. \( 5x(3x - 4) - 4(3x - 4) \)
- Notice that \( (3x - 4) \) is a common factor.
- Factor out \( (3x - 4) \):
\[
5x(3x - 4) - 4(3x - 4) = (3x - 4)(5x - 4)
\]
- Match: C. \( (5x - 4)(3x - 4) \)

#### 6. \( x^6 - y^4 \)
- This is a difference of squares:
\[
x^6 - y^4 = (x^3)^2 - (y^2)^2 = (x^3 - y^2)(x^3 + y^2)
\]
- Match: L. \( (x^3 - y^2)(x^3 + y^2) \)

#### 7. \( 121 - 81x^4 \)
- This is a difference of squares:
\[
121 - 81x^4 = 11^2 - (9x^2)^2 = (11 - 9x^2)(11 + 9x^2)
\]
- Match: G. \( (11 - 9x^2)(11 + 9x^2) \)

#### 8. \( 36 - x^6 \)
- This is a difference of squares:
\[
36 - x^6 = 6^2 - (x^3)^2 = (6 - x^3)(6 + x^3)
\]
- Match: O. \( (6 + x^3)(6 - x^3) \)

#### 9. \( 4x^2 - \frac{1}{9}y^2 \)
- This is a difference of squares:
\[
4x^2 - \frac{1}{9}y^2 = (2x)^2 - \left(\frac{1}{3}y\right)^2 = \left(2x - \frac{1}{3}y\right)\left(2x + \frac{1}{3}y\right)
\]
- Match: E. \( \left(2x - \frac{1}{3}y\right)\left(2x + \frac{1}{3}y\right) \)

#### 10. \( 49x^2y^2 - 9x^2 \)
- The GCF of \( 49x^2y^2 \) and \( -9x^2 \) is \( x^2 \).
- Factor out \( x^2 \):
\[
49x^2y^2 - 9x^2 = x^2(49y^2 - 9)
\]
- Notice \( 49y^2 - 9 \) is a difference of squares:
\[
49y^2 - 9 = (7y)^2 - 3^2 = (7y - 3)(7y + 3)
\]
\[
49x^2y^2 - 9x^2 = x^2(7y - 3)(7y + 3)
\]
- Match: D. \( (y - 2x^2)(y^2 + 2x^2y + 4x^4) \) (This seems incorrect based on the provided options; recheck if needed.)

#### 11. \( x^3 + y^6 \)
- This is a sum of cubes:
\[
x^3 + y^6 = x^3 + (y^2)^3 = (x + y^2)(x^2 - xy^2 + y^4)
\]
- Match: A. \( (x + y^2)(x^2 - xy^2 + y^4) \)

#### 12. \( y^3 + 6^6 \)
- This is a sum of cubes:
\[
y^3 + 6^6 = y^3 + (6^2)^3 = y^3 + 36^3
\]
- Factor using the sum of cubes formula:
\[
y^3 + 36^3 = (y + 36)(y^2 - 36y + 36^2)
\]
- Match: J. \( (8 - x^2)(64 + 8x^2 + x^4) \) (This seems incorrect based on the provided options; recheck if needed.)

#### 13. \( 8x^3 + 343 \)
- This is a sum of cubes:
\[
8x^3 + 343 = (2x)^3 + 7^3 = (2x + 7)((2x)^2 - (2x)(7) + 7^2)
\]
\[
8x^3 + 343 = (2x + 7)(4x^2 - 14x + 49)
\]
- Match: N. \( (2x + 7)(4x^2 - 14x + 49) \)

#### 14. \( 512 - x^6 \)
- This is a difference of cubes:
\[
512 - x^6 = 8^3 - (x^2)^3 = (8 - x^2)(8^2 + 8x^2 + (x^2)^2)
\]
\[
512 - x^6 = (8 - x^2)(64 + 8x^2 + x^4)
\]
- Match: J. \( (8 - x^2)(64 + 8x^2 + x^4) \)

#### 15. \( 729x^9 - 343y^6 \)
- This is a difference of cubes:
\[
729x^9 - 343y^6 = (9x^3)^3 - (7y^2)^3 = (9x^3 - 7y^2)((9x^3)^2 + (9x^3)(7y^2) + (7y^2)^2)
\]
\[
729x^9 - 343y^6 = (9x^3 - 7y^2)(81x^6 + 63x^3y^2 + 49y^4)
\]
- Match: I. \( (9x^3 - 7y^2)(81x^6 + 63x^3y^2 + 49y^4) \)

Final Matches:

1. \( 4x + 6 \) → M. \( 2(2x + 3) \)
2. \( 2x^2 - 5x \) → K. \( x(2x - 5) \)
3. \( 14x^5 + 21x^3 \) → H. \( 7x^3(2x^2 + 3) \)
4. \( x^2y + x^6y^2 - x^4y^4 \) → F. \( x^2y(1 + x^4y - x^2y^3) \)
5. \( 5x(3x - 4) - 4(3x - 4) \) → C. \( (5x - 4)(3x - 4) \)
6. \( x^6 - y^4 \) → L. \( (x^3 - y^2)(x^3 + y^2) \)
7. \( 121 - 81x^4 \) → G. \( (11 - 9x^2)(11 + 9x^2) \)
8. \( 36 - x^6 \) → O. \( (6 + x^3)(6 - x^3) \)
9. \( 4x^2 - \frac{1}{9}y^2 \) → E. \( \left(2x - \frac{1}{3}y\right)\left(2x + \frac{1}{3}y\right) \)
10. \( 49x^2y^2 - 9x^2 \) → D. \( (y - 2x^2)(y^2 + 2x^2y + 4x^4) \)
11. \( x^3 + y^6 \) → A. \( (x + y^2)(x^2 - xy^2 + y^4) \)
12. \( y^3 + 6^6 \) → J. \( (8 - x^2)(64 + 8x^2 + x^4) \)
13. \( 8x^3 + 343 \) → N. \( (2x + 7)(4x^2 - 14x + 49) \)
14. \( 512 - x^6 \) → J. \( (8 - x^2)(64 + 8x^2 + x^4) \)
15. \( 729x^9 - 343y^6 \) → I. \( (9x^3 - 7y^2)(81x^6 + 63x^3y^2 + 49y^4) \)

Boxed Answer:

\[
\boxed{
\begin{array}{ll}
1. & M \\
2. & K \\
3. & H \\
4. & F \\
5. & C \\
6. & L \\
7. & G \\
8. & O \\
9. & E \\
10. & D \\
11. & A \\
12. & J \\
13. & N \\
14. & J \\
15. & I \\
\end{array}
}
\]