Question

Determine the resulting polynomial:
f, of, x, equals, 8, x, minus, 2
f(x)=8x−2
g, of, x, equals, minus, 10, x, squared, minus, x, minus, 7
g(x)=−10x
2
−x−7
Find: , f, of, x, dot, g, of, x
Find: f(x)⋅g(x)

Answer 1

To find the resulting polynomial \( f(x) \cdot g(x) \) where:

- \( f(x) = 8x - 2 \)
- \( g(x) = -10x^2 - x - 7 \)

we will multiply these two polynomials:

\[
f(x) \cdot g(x) = (8x - 2)(-10x^2 - x - 7)
\]

Now we will distribute \( f(x) \) over \( g(x) \):

1. Multiply \( 8x \) by each term in \( g(x) \):
- \( 8x \cdot (-10x^2) = -80x^3 \)
- \( 8x \cdot (-x) = -8x^2 \)
- \( 8x \cdot (-7) = -56x \)

2. Multiply \( -2 \) by each term in \( g(x) \):
- \( -2 \cdot (-10x^2) = 20x^2 \)
- \( -2 \cdot (-x) = 2x \)
- \( -2 \cdot (-7) = 14 \)

Now we combine all these results together:

\[
f(x) \cdot g(x) = -80x^3 + (-8x^2) + (-56x) + 20x^2 + 2x + 14
\]

Next, we can simplify by combining like terms:

1. For the \( x^2 \) terms:
- \( -8x^2 + 20x^2 = 12x^2 \)

2. For the \( x \) terms:
- \( -56x + 2x = -54x \)

Putting it all back together, we have:

\[
f(x) \cdot g(x) = -80x^3 + 12x^2 - 54x + 14
\]

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

\[
\boxed{-80x^3 + 12x^2 - 54x + 14}
\]