Determine the resulting polynomial:

Question

f, of, x, equals, minus, 2, x, minus, 8
f(x)=−2x−8
g, of, x, equals, 4, x, squared, plus, 9, x, minus, 1
g(x)=4x
2
+9x−1
Find: , f, of, x, dot, g, of, x
Find: f(x)⋅g(x)

Answer 1

To find the polynomial \( f(x) \cdot g(x) \), we first need to substitute the given functions into the equation:

Given: \[ f(x) = -2x - 8 \] \[ g(x) = 4x^2 + 9x - 1 \]

Now, we will multiply \( f(x) \) and \( g(x) \):

\[ f(x) \cdot g(x) = (-2x - 8)(4x^2 + 9x - 1) \]

We will distribute \( -2x - 8 \) across \( 4x^2 + 9x - 1 \):

1. Multiply \( -2x \) with each term in \( g(x) \): \[ -2x \cdot 4x^2 = -8x^3 \] \[ -2x \cdot 9x = -18x^2 \] \[ -2x \cdot (-1) = 2x \]

2. Multiply \( -8 \) with each term in \( g(x) \): \[ -8 \cdot 4x^2 = -32x^2 \] \[ -8 \cdot 9x = -72x \] \[ -8 \cdot (-1) = 8 \]

Now, combine all these results together:

\[ f(x) \cdot g(x) = -8x^3 + (-18x^2) + 2x + (-32x^2) + (-72x) + 8 \]

Combine like terms:

1. Combine the \( x^2 \) terms: \[ -18x^2 - 32x^2 = -50x^2 \]

2. Combine the \( x \) terms: \[ 2x - 72x = -70x \]

Putting it all together, we get:

\[ f(x) \cdot g(x) = -8x^3 - 50x^2 - 70x + 8 \]

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

\[ \boxed{-8x^3 - 50x^2 - 70x + 8} \]