Certainly! Let's go through the process of finding [tex]\( f(x) - g(x) \)[/tex] step-by-step.
First, let's review the given functions:
[tex]\[ f(x) = x^3 - 2x^2 + 3x - 5 \][/tex]
[tex]\[ g(x) = x^2 + x - 1 \][/tex]
Our goal is to find [tex]\( f(x) - g(x) \)[/tex]. To achieve this, we subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) - g(x) = (x^3 - 2x^2 + 3x - 5) - (x^2 + x - 1) \][/tex]
Next, we distribute the negative sign through the second polynomial:
[tex]\[ f(x) - g(x) = x^3 - 2x^2 + 3x - 5 - x^2 - x + 1 \][/tex]
Now, we combine like terms:
1. The [tex]\( x^3 \)[/tex] term:
[tex]\[ x^3 \][/tex]
2. The [tex]\( x^2 \)[/tex] terms:
[tex]\[ -2x^2 - x^2 = -3x^2 \][/tex]
3. The [tex]\( x \)[/tex] terms:
[tex]\[ 3x - x = 2x \][/tex]
4. The constant terms:
[tex]\[ -5 + 1 = -4 \][/tex]
Putting it all together, we get:
[tex]\[ f(x) - g(x) = x^3 - 3x^2 + 2x - 4 \][/tex]
Therefore, the result of the function operation [tex]\( f(x) - g(x) \)[/tex] is:
[tex]\[ f(x) - g(x) = x^3 - 3x^2 + 2x - 4 \][/tex]
