Answer :
To solve the given problem, let's investigate the functions [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex] and then their combination. We are given:
[tex]\[ u(x) = x^5 - x^4 + x^2 \][/tex]
[tex]\[ v(x) = -x^2 \][/tex]
We are looking for the expression that matches one of the provided choices. Let's combine [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex]:
[tex]\[ u(x) + v(x) = (x^5 - x^4 + x^2) + (-x^2) \][/tex]
Simplifying this:
[tex]\[ u(x) + v(x) = x^5 - x^4 + x^2 - x^2 \][/tex]
[tex]\[ u(x) + v(x) = x^5 - x^4 \][/tex]
Now, let’s check each given option to see if it matches the simplified expression [tex]\( x^5 - x^4 \)[/tex]:
1. [tex]\( x^3 - x^2 \)[/tex]
2. [tex]\( -x^3 + x^2 \)[/tex]
3. [tex]\( -x^3 + x^2 - 1 \)[/tex]
4. [tex]\( x^3 - x^2 + 1 \)[/tex]
None of these options match [tex]\( x^5 - x^4 \)[/tex].
Since the given options do not match the simplified expression [tex]\( x^5 - x^4 \)[/tex] directly, the conclusion is that none of the given options are equivalent to the combined expression [tex]\( u(x) + v(x) \)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{\text{None of the options are correct}} \][/tex]
[tex]\[ u(x) = x^5 - x^4 + x^2 \][/tex]
[tex]\[ v(x) = -x^2 \][/tex]
We are looking for the expression that matches one of the provided choices. Let's combine [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex]:
[tex]\[ u(x) + v(x) = (x^5 - x^4 + x^2) + (-x^2) \][/tex]
Simplifying this:
[tex]\[ u(x) + v(x) = x^5 - x^4 + x^2 - x^2 \][/tex]
[tex]\[ u(x) + v(x) = x^5 - x^4 \][/tex]
Now, let’s check each given option to see if it matches the simplified expression [tex]\( x^5 - x^4 \)[/tex]:
1. [tex]\( x^3 - x^2 \)[/tex]
2. [tex]\( -x^3 + x^2 \)[/tex]
3. [tex]\( -x^3 + x^2 - 1 \)[/tex]
4. [tex]\( x^3 - x^2 + 1 \)[/tex]
None of these options match [tex]\( x^5 - x^4 \)[/tex].
Since the given options do not match the simplified expression [tex]\( x^5 - x^4 \)[/tex] directly, the conclusion is that none of the given options are equivalent to the combined expression [tex]\( u(x) + v(x) \)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{\text{None of the options are correct}} \][/tex]
