Question

Use the Distributive Property to rewrite the polynomial (x+8)(x−2)(x+8)(x−2) .(2 points) x2+x2+x+x+

Answer 1

To use the Distributive Property to rewrite the polynomial \((x+8)(x−2)(x+8)(x−2)\), we first notice that we can pair the factors:

\[
((x + 8)(x - 2)) \times ((x + 8)(x - 2)).
\]

First, we will simplify \((x + 8)(x - 2)\) using the Distributive Property (also known as the FOIL method for binomials).

\[
(x + 8)(x - 2) = x^2 - 2x + 8x - 16 = x^2 + 6x - 16.
\]

Now, we have:

\[
((x + 8)(x - 2))^2 = (x^2 + 6x - 16)(x^2 + 6x - 16).
\]

Next, we use the Distributive Property again (or the FOIL method) to expand this:

\[
= (x^2 + 6x - 16)(x^2 + 6x - 16).
\]

Using the Distributive Property:

1. \(x^2 \cdot x^2 = x^4\)
2. \(x^2 \cdot 6x = 6x^3\)
3. \(x^2 \cdot -16 = -16x^2\)
4. \(6x \cdot x^2 = 6x^3\)
5. \(6x \cdot 6x = 36x^2\)
6. \(6x \cdot -16 = -96x\)
7. \(-16 \cdot x^2 = -16x^2\)
8. \(-16 \cdot 6x = -96x\)
9. \(-16 \cdot -16 = 256\)

Now, we combine all the terms together:

\[
x^4 + 6x^3 + 6x^3 - 16x^2 + 36x^2 - 96x - 96x + 256.
\]

Combining like terms gives:

\[
x^4 + 12x^3 + (36 - 16 - 16)x^2 - 192x + 256 = x^4 + 12x^3 + 4x^2 - 192x + 256.
\]

Thus, the expanded polynomial form is:

\[
x^4 + 12x^3 + 4x^2 - 192x + 256.
\]

So the final answer is:

\[
x^4 + 12x^3 + 4x^2 - 192x + 256.
\]