Factor completely.
24 y Superscript 6 Baseline minus 2 y Superscript 5 Baseline minus y Superscript 4
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Part 1
24 y Superscript 6 Baseline minus 2 y Superscript 5 Baseline minus y Superscript 4equals
  
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To factor the expression \( 24y^6 - 2y^5 - y^4 \), we follow these steps:

### Step 1: Factor out the greatest common factor (GCF)
The GCF of the terms is \( y^4 \), so we factor that out:

\[
24y^6 - 2y^5 - y^4 = y^4 (24y^2 - 2y - 1)
\]

### Step 2: Factor the quadratic \( 24y^2 - 2y - 1 \)
Next, we factor the quadratic part \( 24y^2 - 2y - 1 \). Using the trial and error method (or factoring by grouping), we look for two numbers that multiply to \( 24 \times (-1) = -24 \) and add to \( -2 \).

The numbers that work are \( -6 \) and \( 4 \), so we split the middle term:

\[
24y^2 - 2y - 1 = 24y^2 - 6y + 4y - 1
\]

Now, group the terms:

\[
= (24y^2 - 6y) + (4y - 1)
\]

Factor each group:

\[
= 6y(4y - 1) + 1(4y - 1)
\]

Finally, factor out the common binomial \( (4y - 1) \):

\[
= (6y + 1)(4y - 1)
\]

### Step 3: Combine the factored parts
Now substitute this factored form back into the original expression:

\[
24y^6 - 2y^5 - y^4 = y^4(6y + 1)(4y - 1)
\]

Thus, the completely factored form is:

\[
\boxed{y^4(6y + 1)(4y - 1)}
\]

Would you like any further details or explanations on this? 

Here are five related questions to explore further:
1. How do you factor more complex quadratics?
2. What are common factoring techniques for polynomials of higher degree?
3. How can factoring help in solving polynomial equations?
4. What is the importance of the greatest common factor in simplifying expressions?
5. How do you check if a polynomial is factored correctly?

**Tip:** Always check your factors by expanding them to ensure you get back the original expression.

Learn how to factor the polynomial 24y^6 - 2y^5 - y^4 by factoring out the greatest common factor (GCF) and using the grouping method for quadratic equations. Suitable for Grades 9-12.

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