To find [tex]\((f+g)(x)\)[/tex] when given [tex]\(f(x) = x^3 - 2x^2 + 1\)[/tex] and [tex]\(g(x) = 4x^3 - 5x + 7\)[/tex], we need to add the two polynomial functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] together.
Step-by-step solution:
1. Write down the given functions:
[tex]\[
f(x) = x^3 - 2x^2 + 1
\][/tex]
[tex]\[
g(x) = 4x^3 - 5x + 7
\][/tex]
2. Add the two functions:
[tex]\[
(f + g)(x) = f(x) + g(x)
\][/tex]
Substituting the given expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f + g)(x) = (x^3 - 2x^2 + 1) + (4x^3 - 5x + 7)
\][/tex]
3. Combine like terms:
Start by adding the [tex]\(x^3\)[/tex] terms:
[tex]\[
x^3 + 4x^3 = 5x^3
\][/tex]
Add the [tex]\(x^2\)[/tex] terms (note that there is no [tex]\(x^2\)[/tex] term in [tex]\(g(x)\)[/tex]):
[tex]\[
-2x^2
\][/tex]
Add the [tex]\(x\)[/tex] terms (note that there is no [tex]\(x\)[/tex] term in [tex]\(f(x)\)[/tex]):
[tex]\[
-5x
\][/tex]
Add the constant terms:
[tex]\[
1 + 7 = 8
\][/tex]
4. Combine all parts together :
[tex]\[
(f + g)(x) = 5x^3 - 2x^2 - 5x + 8
\][/tex]
Therefore, the resulting polynomial for [tex]\((f+g)(x)\)[/tex] is:
[tex]\[
(f+g)(x) = 5x^3 - 2x^2 - 5x + 8
\][/tex]
