Question

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?
options: 8, 32, 4, 16

Explanation:

Step1: Apply the distributive property (FOIL for polynomials)

Multiply each term in the first polynomial by each term in the second polynomial:
$$(y - 4x)(y^2 + 4y + 16) = y(y^2 + 4y + 16) - 4x(y^2 + 4y + 16)$$

Step2: Distribute further

For the first part: \(y \cdot y^2 + y \cdot 4y + y \cdot 16 = y^3 + 4y^2 + 16y\)
For the second part: \(-4x \cdot y^2 - 4x \cdot 4y - 4x \cdot 16 = -4xy^2 - 16xy - 64x\)

Step3: Combine the two parts

The product is \(y^3 + 4y^2 + 16y - 4xy^2 - 16xy - 64x\)

Step4: Compare with the given polynomial form

The given form is \(y^3 + 4y^2 + ay - 4xy^2 - axy - 64x\). By comparing the coefficients of \(y\) and \(xy\), we see that \(a = 16\) (since the coefficient of \(y\) is \(16\) and the coefficient of \(xy\) is \(-16\), so \(a = 16\)). Wait, no, wait. Wait, in the second part, when we distribute \(-4x\) to \(4y\), it's \(-4x \times 4y = -16xy\), and when we distribute \(y\) to \(16\), it's \(16y\). So in the given polynomial, the term with \(y\) is \(ay\) and the term with \(xy\) is \(-axy\). So comparing \(16y\) with \(ay\) and \(-16xy\) with \(-axy\), we get \(a = 16\)? Wait, no, wait the options have 16 as an option. Wait, let's check again.

Wait, let's redo the distribution:

First, multiply \(y\) by each term in the second polynomial:

\(y \times y^2 = y^3\)

\(y \times 4y = 4y^2\)

\(y \times 16 = 16y\)

Then, multiply \(-4x\) by each term in the second polynomial:

\(-4x \times y^2 = -4xy^2\)

\(-4x \times 4y = -16xy\)

\(-4x \times 16 = -64x\)

So combining all terms: \(y^3 + 4y^2 + 16y - 4xy^2 - 16xy - 64x\)

Now, the given polynomial is \(y^3 + 4y^2 + ay - 4xy^2 - axy - 64x\). So for the term \(ay\), that's \(16y\), so \(a = 16\). For the term \(-axy\), that's \(-16xy\), so \(a = 16\). So the value of \(a\) is 16. Wait, but the options include 16? Wait, the options shown are 8, 32, 4, 16? Wait, the user's image shows a box with 16? Wait, the user's image: the options are 8, 32, 4, and 16 (the bottom left box). So yes, \(a = 16\).

Wait, but let's check again. Wait, maybe I made a mistake. Wait, the original problem: \((y - 4x)(y^2 + 4y + 16)\). Let's recall that \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Here, \(y^3 - (4x)^3 = (y - 4x)(y^2 + 4xy + 16x^2)\)? Wait, no, wait the second polynomial is \(y^2 + 4y + 16\), not \(y^2 + 4xy + 16x^2\). Wait, that's a mistake. Wait, no, the problem is as given: \((y - 4x)(y^2 + 4y + 16)\). So my initial distribution is correct. So when we multiply \(y\) by \(16\), we get \(16y\), and when we multiply \(-4x\) by \(4y\), we get \(-16xy\). So comparing to the given form \(y^3 + 4y^2 + ay - 4xy^2 - axy - 64x\), so \(ay = 16y\) implies \(a = 16\), and \(-axy = -16xy\) implies \(a = 16\). So \(a = 16\).

Wait, but let's check the options. The options are 8, 32, 4, 16 (the bottom left box). So the correct answer is 16.

Answer:

16 (the option with 16, likely the bottom left box in the image)