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### Review and Preview to Chapter 1

#### 1. Factor Fully:
a) \( x^2 - 49 \)
- This is a difference of squares: 
  \[ (x - 7)(x + 7) \]

b) \( y^2 + 4y + 4 \)
- This is a perfect square trinomial: 
  \[ (y + 2)(y + 2) \text{ or } (y + 2)^2 \]

c) \( 3x^2 + 10x + 8 \)
- Factor the trinomial:
  \[ (3x + 4)(x + 2) \]

d) \( 9y^2 - 16 \)
- This is another difference of squares:
  \[ (3y - 4)(3y + 4) \]

#### 2. Factor Fully:
a) \( x^2 + 6x - 16 \)
- Find factors of -16 that add to 6: None exist, this is not factorable over integers.

b) \( 2x^2 + 3x - 9 \)
- Factor the trinomial:
  \[ (2x - 3)(x + 3) \]

c) \( 6x^2 - 4x + 10x - 8 \)
- Combine terms and factor:
  \[ 2x(3x - 2) + 4(3x - 2) = (3x - 2)(2x + 4) \]

d) \( 9a^2 - b^2 \)
- This is a difference of squares:
  \[ (3a - b)(3a + b) \]

#### 3. Factor Fully:
a) \( a^2 + ab - 6b^2 \)
- Factor the trinomial:
  \[ (a + 3b)(a - 2b) \]

b) \( 4p^2 + 4pq + q^2 \)
- This is a perfect square trinomial:
  \[ (2p + q)^2 \]

c) \( 4p^2 - 25q^2 \)
- This is a difference of squares:
  \[ (2p - 5q)(2p + 5q) \]

#### 4. Factor Fully:
a) \( 2x^2 - 10x \)
- Factor out the greatest common factor (GCF):
  \[ 2x(x - 5) \]

b) \( x^3 - 3x^2 - 4x + 12 \)
- Factor by grouping:
  \[ x^2(x - 3) - 4(x - 3) = (x - 3)(x^2 - 4) = (x - 3)(x - 2)(x + 2) \]

c) \( 5y^3 - 20 \)
- Factor out the GCF:
  \[ 5(y^3 - 4) \]

#### 5. Factor Fully:
a) \( x^3 + 2x^2 - 5x - 10 \)
- Factor by grouping:
  \[ x^2(x + 2) - 5(x + 2) = (x + 2)(x^2 - 5) \]

b) \( 4m^2 - 4mn + n^2 \)
- This is a perfect square trinomial:
  \[ (2m - n)^2 \]

c) \( ab^2 + a^2b + b^3 \)
- Factor out the GCF:
  \[ b(ab + a^2 + b^2) \]

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### Chapter 6 Review Exercise

#### Fully Factor the Following:
a) \( x^4 + 3x^3 + 3x^2 + x \)
- Factor by grouping:
  \[ x(x^3 + 3x^2 + 3x + 1) = x(x + 1)^3 \]

b) \( x^4 - 25x^2 \)
- Factor the difference of squares:
  \[ x^2(x^2 - 25) = x^2(x - 5)(x + 5) \]

c) \( y^3 - 8 \)
- This is a difference of cubes:
  \[ (y - 2)(y^2 + 2y + 4) \]

d) \( x^4 - 25x^2 \)
- This was already factored above:
  \[ x^2(x - 5)(x + 5) \]

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### Sum and Difference of Cubes

#### 1. Factor:
a) \( b^3 + 27 \)
- This is a sum of cubes:
  \[ (b + 3)(b^2 - 3b + 9) \]

b) \( a^3 + 125 \)
- This is a sum of cubes:
  \[ (a + 5)(a^2 - 5a + 25) \]

#### 2. Factor:
a) \( y^3 - 125 \)
- This is a difference of cubes:
  \[ (y - 5)(y^2 + 5y + 25) \]

b) \( z^3 + 1 \)
- This is a sum of cubes:
  \[ (z + 1)(z^2 - z + 1) \]

#### 3. Factor:
a) \( x^3 + 8 \)
- This is a sum of cubes:
  \[ (x + 2)(x^2 - 2x + 4) \]

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This covers the entire problem set from the image. Let me know if you want further clarifications or a more detailed breakdown on any of these solutions.

### Related Questions:
1. How do you determine when to use difference of squares for factoring?
2. Can all trinomials be factored over integers? Why or why not?
3. What is the key difference between factoring sums of cubes and differences of cubes?
4. How do you recognize a perfect square trinomial?
5. What are the main strategies to factor polynomials by grouping?

### Tip:
Always check for a greatest common factor (GCF) before attempting more advanced factoring techniques; it simplifies the process significantly!

Solve all of the questions in the uploaded image

This page provides detailed solutions to polynomial factoring problems, covering concepts such as difference of squares, sum and difference of cubes, and perfect square trinomials. Suitable for students in Grades 9-12.

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