Determine the resulting polynomial:

Question

f, of, x, equals, 2, x, plus, 6
f(x)=2x+6
g, of, x, equals, 6, x, squared, minus, x, plus, 10
g(x)=6x
2
−x+10
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

\[ f(x) = 2x + 6 \]

and

\[ g(x) = 6x^2 - x + 10, \]

we will multiply these two polynomials together.

Step 1: Set up the multiplication

First, we write down the expression:

\[ f(x) \cdot g(x) = (2x + 6)(6x^2 - x + 10) \]

Step 2: Distribute each term in \(f(x)\)

We will distribute each term in \( f(x) \) over \( g(x) \):

1. Multiply \( 2x \) by each term in \( g(x) \):

   • \( 2x \cdot 6x^2 = 12x^3 \)
   • \( 2x \cdot (-x) = -2x^2 \)
   • \( 2x \cdot 10 = 20x \)

2. Multiply \( 6 \) by each term in \( g(x) \):

   • \( 6 \cdot 6x^2 = 36x^2 \)
   • \( 6 \cdot (-x) = -6x \)
   • \( 6 \cdot 10 = 60 \)

Step 3: Combine the results

Now, we combine all these results:

\[ 12x^3 + (-2x^2) + 20x + 36x^2 + (-6x) + 60 \]

Step 4: Combine like terms

Next, we combine the like terms:

• The \( x^3 \) term: \( 12x^3 \)
• The \( x^2 \) terms: \( -2x^2 + 36x^2 = 34x^2 \)
• The \( x \) terms: \( 20x - 6x = 14x \)
• The constant term: \( 60 \)

Resulting Polynomial

So, the resulting polynomial is:

\[ f(x) \cdot g(x) = 12x^3 + 34x^2 + 14x + 60 \]

Thus, the answer is:

\[ \boxed{12x^3 + 34x^2 + 14x + 60} \]