Using the distributive property to find the product (*y*−4*x*)(*y*2+4*y*+16) results in a polynomial of the form *y*3+4*y*2+*ay*−4*xy*2−*axy*−64*x*. What is the value of *a* in the polynomial?

4

8

16

32

To find the value of \( a \), we need to apply the distributive property to multiply the two polynomials \( (y - 4x)(y^2 + 4y + 16) \).

Let's perform the multiplication step by step.

### Step 1: Multiply each term in \( (y - 4x) \) by each term in \( (y^2 + 4y + 16) \).

1. Multiply \( y \) by each term in \( (y^2 + 4y + 16) \):
   \[
   y \cdot y^2 = y^3
   \]
   \[
   y \cdot 4y = 4y^2
   \]
   \[
   y \cdot 16 = 16y
   \]

2. Multiply \( -4x \) by each term in \( (y^2 + 4y + 16) \):
   \[
   -4x \cdot y^2 = -4xy^2
   \]
   \[
   -4x \cdot 4y = -16xy
   \]
   \[
   -4x \cdot 16 = -64x
   \]

### Step 2: Combine all the terms.

Now, let's put all the results together:
\[
y^3 + 4y^2 + 16y - 4xy^2 - 16xy - 64x
\]

The polynomial we obtained matches the form given in the problem:
\[
y^3 + 4y^2 + ay - 4xy^2 - axy - 64x
\]

### Step 3: Compare the terms.

By comparing the like terms, we see that:
\[
16y = ay \quad \text{and} \quad -16xy = -axy
\]

Thus, \( a = 16 \).

### Final Answer:
The value of \( a \) is **16**.

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Would you like further details or have any questions?

Here are five related questions to explore further:
1. How does the distributive property work with polynomials?
2. What are the key steps in multiplying two binomials?
3. Can you generalize how to distribute a monomial over a trinomial?
4. How would you identify like terms in polynomial expressions?
5. What role does the distributive property play in factoring polynomials?

**Tip:** Always group and organize like terms after distributing, as this helps simplify the polynomial expression efficiently.

Using the distributive property to find the product (y−4x)(y^2+4y+16) results in a polynomial of the form y^3+4y^2+ay−4xy^2−axy−64x. What is the value of a in the polynomial? 4, 8, 16, 32

This problem demonstrates how to use the distributive property to multiply two polynomials and find the value of 'a' in the resulting expression. Learn how to break down the multiplication and identify like terms, suitable for Grades 9-10.

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