Factor completely.
15 y Superscript 5 Baseline plus 7 y Superscript 4 Baseline minus 10 y cubed
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.


Let's factor the expression \( 15y^5 + 7y^4 - 10y^3 \) step by step.

### Step 1: Factor out the greatest common factor (GCF)
The GCF of the terms \( 15y^5 \), \( 7y^4 \), and \( 10y^3 \) is \( y^3 \). So, we first factor out \( y^3 \):

\[
y^3(15y^2 + 7y - 10)
\]

### Step 2: Factor the quadratic expression
Now, we need to factor the quadratic expression \( 15y^2 + 7y - 10 \). We'll use the **ac method** for factoring.

1. Multiply \( a = 15 \) (coefficient of \( y^2 \)) and \( c = -10 \) (constant term):
   \[
   15 \times (-10) = -150
   \]
2. Find two numbers that multiply to \( -150 \) and add up to \( 7 \) (coefficient of \( y \)). These numbers are \( 15 \) and \( -10 \), because:
   \[
   15 \times (-10) = -150 \quad \text{and} \quad 15 + (-10) = 7
   \]

3. Rewrite the middle term using these numbers:
   \[
   15y^2 + 15y - 10y - 10
   \]

4. Factor by grouping:
   \[
   (15y^2 + 15y) - (10y + 10)
   \]
   Factor out the GCF from each group:
   \[
   15y(y + 1) - 10(y + 1)
   \]
5. Factor out the common binomial factor \( (y + 1) \):
   \[
   (15y - 10)(y + 1)
   \]
6. Simplify \( (15y - 10) \) by factoring out the GCF of 5:
   \[
   5(3y - 2)(y + 1)
   \]

### Step 3: Final factored form
Thus, the completely factored form of \( 15y^5 + 7y^4 - 10y^3 \) is:

\[
y^3 \cdot 5(3y - 2)(y + 1) = 5y^3(3y - 2)(y + 1)
\]

### Final Answer:
\[
\boxed{5y^3(3y - 2)(y + 1)}
\]

Would you like further details on any of the steps?

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Here are 5 related questions:
1. How do you factor a quadratic trinomial using the ac method?
2. What is the greatest common factor (GCF), and how do you find it?
3. How do you factor by grouping?
4. What is the difference between factoring and expanding a polynomial?
5. How can you check if your factored form is correct?

**Tip:** Always check your factorization by expanding the terms back to ensure correctness.

Learn how to factor the polynomial 15y^5 + 7y^4 - 10y^3 step-by-step. This solution covers factoring polynomials using the GCF and factoring by grouping techniques, suitable for students in Grades 9-12.

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