Question

Determine the resulting polynomial:
f, of, x, equals, minus, x, minus, 1
f(x)=−x−1
g, of, x, equals, minus, 10, x, squared, plus, x, minus, 6
g(x)=−10x
2
+x−6
Find: , f, of, x, dot, g, of, x
Find: f(x)⋅g(x)

Answer 1

To find \( f(x) \cdot g(x) \), we first need to substitute the given functions into the expression.

We know:
\[
f(x) = -x - 1
\]
\[
g(x) = -10x^2 + x - 6
\]

Now, we will compute \( f(x) \cdot g(x) \):
\[
f(x) \cdot g(x) = (-x - 1) \cdot (-10x^2 + x - 6)
\]

Now let's perform the multiplication by distributing \( -x - 1 \) to each term in \( g(x) \):

1. First, multiply \( -x \) with each term in \( g(x) \):
\[
-x \cdot (-10x^2) = 10x^3
\]
\[
-x \cdot x = -x^2
\]
\[
-x \cdot (-6) = 6x
\]

So, from this part, we get:
\[
10x^3 - x^2 + 6x
\]

2. Next, multiply \( -1 \) with each term in \( g(x) \):
\[
-1 \cdot (-10x^2) = 10x^2
\]
\[
-1 \cdot x = -x
\]
\[
-1 \cdot (-6) = 6
\]

So, from this part, we get:
\[
10x^2 - x + 6
\]

Now, we combine all the terms obtained from Steps 1 and 2:
\[
f(x) \cdot g(x) = (10x^3 - x^2 + 6x) + (10x^2 - x + 6)
\]

Combining like terms:
- The \( x^3 \) term: \( 10x^3 \)
- The \( x^2 \) terms: \( -x^2 + 10x^2 = 9x^2 \)
- The \( x \) terms: \( 6x - x = 5x \)
- The constant term: \( 6 \)

Thus, the resulting polynomial \( f \cdot g \) is:
\[
f(x) \cdot g(x) = 10x^3 + 9x^2 + 5x + 6
\]

Therefore, the final answer is:
\[
\boxed{10x^3 + 9x^2 + 5x + 6}
\]