To simplify the expression [tex]\(\left(-7 z^2+3 z^4+7\right)+\left(-5 z-9 z^4\right)\)[/tex], we need to follow a series of steps to combine like terms. Let’s go through them one by one:
1. Write out the entire expression:
[tex]\[
(-7 z^2 + 3 z^4 + 7) + (-5 z - 9 z^4)
\][/tex]
2. Combine like terms (terms with the same power of [tex]\(z\)[/tex]):
- Look at the [tex]\(z^4\)[/tex] terms:
[tex]\[
3 z^4 - 9 z^4
\][/tex]
Simplified, this is:
[tex]\[
-6 z^4
\][/tex]
- Look at the [tex]\(z^2\)[/tex] terms:
[tex]\[
-7 z^2
\][/tex]
There are no other [tex]\(z^2\)[/tex] terms to combine with, so this stays:
[tex]\[
-7 z^2
\][/tex]
- Look at the [tex]\(z\)[/tex] terms:
[tex]\[
-5 z
\][/tex]
There are no other [tex]\(z\)[/tex] terms to combine with, so this stays:
[tex]\[
-5 z
\][/tex]
- Look at the constant terms:
[tex]\[
7
\][/tex]
There are no other constant terms to combine with, so this stays:
[tex]\[
7
\][/tex]
3. Combine all the simplified terms:
[tex]\[
-6 z^4 - 7 z^2 - 5 z + 7
\][/tex]
By combining all the like terms correctly, we get the simplified expression:
[tex]\[
\boxed{-6 z^4 - 7 z^2 - 5 z + 7}
\][/tex]
We compare this simplified expression with the given choices and find that none exactly match. Let’s double-check our original options though:
(A) [tex]\(-6 z^4 - 12 z^2 + 7\)[/tex]
(B) [tex]\(-16 z^4 - 2 z^2 + 7\)[/tex]
(C) [tex]\(-6 z^6 - 7 z^2 - 5 z + 7\)[/tex]
(D) [tex]\(-16 z^4 - 7 z^2 - 5 z + 7\)[/tex]
None match directly but considering our simplified expression [tex]\(\boxed{-6 z^4 - 7 z^2 - 5 z + 7}\)[/tex], it is evident there seems to be a problem in choices given maybe an option error thus best possible verified ans is directly computed above manually as `well simplified expression`:
So the correct simplified expression is:
[tex]\[
\boxed{-6 z^4 - 7 z^2 - 5 z + 7}
\][/tex]
