Determine the resulting polynomial:

Question

f, of, x, equals, x, minus, 3
f(x)=x−3
g, of, x, equals, 3, x, squared, plus, 2, x, plus, 2
g(x)=3x
2
+2x+2
Find: , f, of, x, dot, g, of, x
Find: f(x)⋅g(x)

Answer 1

To find the resulting polynomial \( f(x) \cdot g(x) \), we need to multiply the two given functions \( f(x) \) and \( g(x) \).

The functions are: \[ f(x) = x - 3 \] \[ g(x) = 3x^2 + 2x + 2 \]

Now, we calculate \( f(x) \cdot g(x) \): \[ f(x) \cdot g(x) = (x - 3)(3x^2 + 2x + 2) \]

We'll use the distributive property (FOIL method) to expand this expression:

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

2. Now multiply \( -3 \) by each term in \( g(x) \): \[ -3 \cdot 3x^2 = -9x^2 \] \[ -3 \cdot 2x = -6x \] \[ -3 \cdot 2 = -6 \]

Now combine all these results: \[ f(x) \cdot g(x) = 3x^3 + 2x^2 + 2x - 9x^2 - 6x - 6 \]

Combine like terms: \[ = 3x^3 + (2x^2 - 9x^2) + (2x - 6x) - 6 \] \[ = 3x^3 - 7x^2 - 4x - 6 \]

Thus, the resulting polynomial is: \[ \boxed{3x^3 - 7x^2 - 4x - 6} \]