Factor the following product
1. y³-69=(y-9) (y+9y+16y)
2.125y³-8=(5y-2)
3. 125x³-y³=
4.125x³-8y³=
For example: the product is x³-y³ and the factor us (x-y) (x²-xy+y²)

Let's factor each of the given expressions step by step.

### 1. \( y^3 - 69 = (y - 9)(y + 9y + 16y) \)

This doesn't appear to be a correctly structured equation, and the right-hand side is likely incorrect. Let's correct it first.

Assuming you meant to factor \( y^3 - 69 \), which is not easily factored over the real numbers unless we use complex roots or numerical approximations. Since \( y^3 - 69 \) is not a difference of cubes or squares, it's generally left in its simplest form unless specific factors are given.

**Possible Correct Interpretation**: If you wanted to use a cubic identity like \( y^3 - a^3 \), we would factor it using:

\[
y^3 - a^3 = (y - a)(y^2 + ay + a^2)
\]

But since \( 69 \) is not a perfect cube, this isn't applicable here. 

However, let's correct the expression first before proceeding if there's more context.

### 2. \( 125y^3 - 8 = (5y - 2) \)

This is a difference of cubes:

\[
125y^3 - 8 = (5y)^3 - 2^3
\]

Using the difference of cubes identity:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Let \( a = 5y \) and \( b = 2 \):

\[
125y^3 - 8 = (5y - 2)\left((5y)^2 + (5y)(2) + 2^2\right)
\]

\[
= (5y - 2)(25y^2 + 10y + 4)
\]

### 3. \( 125x^3 - y^3 \)

Again, this is a difference of cubes:

\[
125x^3 - y^3 = (5x)^3 - y^3
\]

Using the difference of cubes identity:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Let \( a = 5x \) and \( b = y \):

\[
125x^3 - y^3 = (5x - y)(25x^2 + 5xy + y^2)
\]

### 4. \( 125x^3 - 8y^3 \)

This is a difference of cubes as well:

\[
125x^3 - 8y^3 = (5x)^3 - (2y)^3
\]

Using the difference of cubes identity:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Let \( a = 5x \) and \( b = 2y \):

\[
125x^3 - 8y^3 = (5x - 2y)(25x^2 + 10xy + 4y^2)
\]

### Summary of Factorizations:

1. \( y^3 - 69 \): Needs clarification or correction.
2. \( 125y^3 - 8 = (5y - 2)(25y^2 + 10y + 4) \)
3. \( 125x^3 - y^3 = (5x - y)(25x^2 + 5xy + y^2) \)
4. \( 125x^3 - 8y^3 = (5x - 2y)(25x^2 + 10xy + 4y^2) \)

Would you like more details on any specific step? Or do you need further help?

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Here are five related questions for practice:

1. Factor \( 64x^3 - 1 \).
2. Factor \( 27x^3 + 8y^3 \).
3. Factor \( 8x^3 - 125y^3 \).
4. Factor \( x^3 + 8 \).
5. Factor \( 216x^3 - 27y^3 \).

**Tip**: When factoring cubes, always check if the expression is a sum or difference of cubes and apply the respective formula.

Learn how to factor cubic expressions like y^3 - 69, 125y^3 - 8, 125x^3 - y^3, and 125x^3 - 8y^3 with step-by-step solutions. Understand the difference of cubes formula and its applications. Suitable for high school students studying algebra and cubic equations.

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