Question

Determine the resulting polynomial:
f, of, x, equals, minus, 2, x, minus, 2
f(x)=−2x−2
g, of, x, equals, 8, x, squared, minus, x, minus, 4
g(x)=8x
2
−x−4
Find: , f, of, x, dot, g, of, x
Find: f(x)⋅g(x)

Answer 1

To find the product of the two polynomials \( f(x) \) and \( g(x) \), we start with the given functions:

\[
f(x) = -2x - 2
\]
\[
g(x) = 8x^2 - x - 4
\]

Now we will calculate \( f(x) \cdot g(x) \):

\[
f(x) \cdot g(x) = (-2x - 2)(8x^2 - x - 4)
\]

Next, we need to distribute \( f(x) \) over \( g(x) \):

1. **Distributing \(-2x\)**:
\[
-2x \cdot 8x^2 = -16x^3
\]
\[
-2x \cdot (-x) = 2x^2
\]
\[
-2x \cdot (-4) = 8x
\]

2. **Distributing \(-2\)**:
\[
-2 \cdot 8x^2 = -16x^2
\]
\[
-2 \cdot (-x) = 2x
\]
\[
-2 \cdot (-4) = 8
\]

Now we can combine all these terms:

\[
f(x) \cdot g(x) = -16x^3 + 2x^2 + 8x - 16x^2 + 2x + 8
\]

Combining like terms:

\[
= -16x^3 + (2x^2 - 16x^2) + (8x + 2x) + 8
\]
\[
= -16x^3 - 14x^2 + 10x + 8
\]

Thus, the resulting polynomial \( f(x) \cdot g(x) \) is:

\[
\boxed{-16x^3 - 14x^2 + 10x + 8}
\]