Question

Determine the resulting polynomial:
f, of, x, equals, minus, x, minus, 10
f(x)=−x−10
g, of, x, equals, minus, 10, x, squared, plus, 5, x, minus, 4
g(x)=−10x
2
+5x−4
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 know the explicit forms of the functions \( f(x) \) and \( g(x) \).

Given:

\[
f(x) = -x - 10
\]
\[
g(x) = -10x^2 + 5x - 4
\]

Now, we multiply these two polynomials together:

\[
f(x) \cdot g(x) = (-x - 10)(-10x^2 + 5x - 4)
\]

We can distribute \( -x - 10 \) to each term in \( g(x) \):

1. **Multiply \( -x \) by each term in \( g(x) \)**:
\[
-x \cdot -10x^2 = 10x^3
\]
\[
-x \cdot 5x = -5x^2
\]
\[
-x \cdot -4 = 4x
\]

So, this yields:
\[
10x^3 - 5x^2 + 4x
\]

2. **Multiply \( -10 \) by each term in \( g(x) \)**:
\[
-10 \cdot -10x^2 = 100x^2
\]
\[
-10 \cdot 5x = -50x
\]
\[
-10 \cdot -4 = 40
\]

So, this yields:
\[
100x^2 - 50x + 40
\]

3. **Now, combine all these results**:
\[
f(x) \cdot g(x) = (10x^3 - 5x^2 + 4x) + (100x^2 - 50x + 40)
\]

4. **Combine like terms**:
\[
= 10x^3 + (-5x^2 + 100x^2) + (4x - 50x) + 40
\]
\[
= 10x^3 + 95x^2 - 46x + 40
\]

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

\[
\boxed{10x^3 + 95x^2 - 46x + 40}
\]