To find [tex]\( f(x) + g(x) \)[/tex] and its degree, we need to add the corresponding coefficients of the polynomials [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. Let's break down each polynomial term by term.
Given:
[tex]\[ f(x) = 3x^5 - 4x^3 + ax + x - 7 \][/tex]
[tex]\[ g(x) = -3x^5 + 4x^3 + 6x^2 - x + 7 \][/tex]
First, we align the terms with the same powers:
[tex]\[
\begin{aligned}
f(x) &= 3x^5 + 0x^4 - 4x^3 + 0x^2 + (a+1)x - 7, \\
g(x) &= -3x^5 + 0x^4 + 4x^3 + 6x^2 - x + 7.
\end{aligned}
\][/tex]
Next, we add the corresponding coefficients:
1. Coefficient of [tex]\( x^5 \)[/tex]:
[tex]\[
3 + (-3) = 0
\][/tex]
2. Coefficient of [tex]\( x^4 \)[/tex]:
[tex]\[
0 + 0 = 0
\][/tex]
3. Coefficient of [tex]\( x^3 \)[/tex]:
[tex]\[
-4 + 4 = 0
\][/tex]
4. Coefficient of [tex]\( x^2 \)[/tex]:
[tex]\[
0 + 6 = 6
\][/tex]
5. Coefficient of [tex]\( x \)[/tex]:
[tex]\[
(a + 1) - 1 = a
\][/tex]
6. Constant term:
[tex]\[
-7 + 7 = 0
\][/tex]
Therefore, the resulting polynomial [tex]\( f(x) + g(x) \)[/tex] is:
[tex]\[ f(x) + g(x) = 6x^2 + ax \][/tex]
This polynomial can be rewritten as [tex]\( 0x^5 + 0x^4 + 0x^3 + 6x^2 + ax + 0 \)[/tex], making it clear that the highest non-zero coefficient corresponds to the [tex]\( x^2 \)[/tex] term.
The degree of the polynomial [tex]\( f(x) + g(x) \)[/tex] is determined by the highest power of [tex]\( x \)[/tex] with a non-zero coefficient. Thus, the degree of [tex]\( f(x) + g(x) \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
