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It seems you're working on dividing polynomials using long division, specifically dividing \( y - 5 \) into \( y^6 - 1 \).

Let's break this down step by step:

### Step 1: Set up the division
You're trying to divide \( y^6 - 1 \) by \( y - 5 \).

### Step 2: First division
The first term of \( y^6 \) divided by \( y \) gives:
\[
y^5
\]
Now multiply \( y^5 \) by \( y - 5 \):
\[
y^5 \times (y - 5) = y^6 - 5y^5
\]
Subtract this result from \( y^6 - 1 \):
\[
(y^6 - 1) - (y^6 - 5y^5) = 5y^5 - 1
\]

### Step 3: Second division
Now divide \( 5y^5 \) by \( y \), which gives:
\[
5y^4
\]
Multiply \( 5y^4 \) by \( y - 5 \):
\[
5y^4 \times (y - 5) = 5y^5 - 25y^4
\]
Subtract this from \( 5y^5 - 1 \):
\[
(5y^5 - 1) - (5y^5 - 25y^4) = 25y^4 - 1
\]

### Step 4: Continue the process
Keep dividing each resulting term by \( y \) and subtracting until you've completed the long division.

Would you like further details on the next steps? 

Here are five questions to consider:
1. What is the next term in the quotient after \( 5y^4 \)?
2. How do you handle the remainder in polynomial long division?
3. How does polynomial long division relate to synthetic division?
4. Can this division problem be factored differently for an easier solution?
5. What are the applications of polynomial long division in higher mathematics?

**Tip:** Always align terms of the same degree while performing polynomial division to keep the process organized!

Dividing Polynomials using long division: y - 5 into y^6 - 1

Learn how to divide the polynomial y^6 - 1 by y - 5 using long division. This step-by-step solution covers polynomial division and algebraic concepts. Ideal for students in Grades 10-12.

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