https://file.aiquickdraw.com/mathbot/file/172764930642080reysjr.jpg

The expression given in the image is:

\[
y^8 - 8y^7 + y - 8
\]

To factor this expression, we'll start by observing the terms. This is a cubic-like structure with alternating powers of \( y \) and constant terms. Let's attempt factoring it as a grouping or a substitution. One useful approach is to notice a similarity with a quadratic form.

### Step-by-Step Factorization:

1. **Substitute**: Let’s make a substitution to simplify the powers of \( y \). Set \( z = y^7 \), so we rewrite the expression:
   \[
   y^8 - 8y^7 + y - 8 = y \cdot z - 8z + y - 8
   \]
   This gives us:
   \[
   y(z - 8) + (y - 8)
   \]

2. **Factor by grouping**: Now, observe that both terms contain \( (z - 8) \), so we can factor it out:
   \[
   = (y + 1)(y^7 - 8)
   \]

3. Therefore, the factored form is:
   \[
   (y + 1)(y^7 - 8)
   \]

This is the factorized form of the given expression.

Would you like further clarification or details? Here are some related questions to deepen your understanding:

1. How do you recognize patterns in expressions to use substitution effectively?
2. Could this factorization approach work on other types of polynomial expressions?
3. What is the significance of factoring in simplifying complex equations?
4. How do you know when to factor by grouping versus other factoring methods?
5. How could the factorization of higher-degree polynomials be approached differently?

**Tip**: Whenever dealing with higher-degree polynomials, always look for substitution or grouping methods that simplify the powers, especially when there are common patterns between terms.

Factor the following expression: y^8 - 8y^7 + y - 8

Learn how to factor the polynomial expression y^8 - 8y^7 + y - 8. This problem focuses on algebraic techniques like grouping and substitution to simplify higher-degree polynomials. Ideal for students in Grades 9-12.

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