Let's start by examining the given functions:
[tex]\[ f(x) = x^3 + x^2 + 1 \][/tex]
[tex]\[ g(x) = -6x^2 + 2 \][/tex]
To find the sum of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we add the expressions together:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substituting the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f+g)(x) = (x^3 + x^2 + 1) + (-6x^2 + 2) \][/tex]
Combine like terms (terms with the same power of [tex]\( x \)[/tex]):
[tex]\[ (f+g)(x) = x^3 + x^2 - 6x^2 + 1 + 2 \][/tex]
[tex]\[ (f+g)(x) = x^3 - 5x^2 + 3 \][/tex]
So, the polynomial that represents [tex]\( (f+g)(x) \)[/tex] is:
[tex]\[ x^3 - 5x^2 + 3 \][/tex]
Now, we need to match this with one of the given options:
1. [tex]\( x^3 + 7x^2 + 3 \)[/tex]
2. [tex]\( x^3 - 7x^2 + 5 \)[/tex]
3. [tex]\( x^3 + 5x^2 + 3 \)[/tex]
4. [tex]\( x^3 - 5x^2 + 3 \)[/tex]
Comparing our derived polynomial [tex]\( x^3 - 5x^2 + 3 \)[/tex] with the options, we see that it matches option 4:
[tex]\[ x^3 - 5x^2 + 3 \][/tex]
Thus, the correct polynomial for [tex]\( (f+g)(x) \)[/tex] is:
[tex]\[ x^3 - 5x^2 + 3 \][/tex]
In terms of the domain, since both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are polynomials, their sum [tex]\( (f+g)(x) \)[/tex] is also a polynomial. Polynomials are defined for all real numbers, thus:
[tex]\[ \text{The domain of } (f+g)(x) \text{ is all real numbers.} \][/tex]
So, the steps are:
1. Combine like terms of the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex].
2. Determine the resulting polynomial.
3. Compare the result with the provided options.
4. State the domain (which is all real numbers for polynomials).
The final answer is:
[tex]\[ x^3 - 5x^2 + 3 \][/tex]
