Answer :
To solve the expression and find which of the given choices is equivalent to [tex]\(-4x^3 - 12x^3 + 9x^2\)[/tex], let's simplify it step-by-step:
1. Identify Like Terms:
- The terms [tex]\(-4x^3\)[/tex] and [tex]\(-12x^3\)[/tex] are like terms because they both have [tex]\(x^3\)[/tex].
- The term [tex]\(9x^2\)[/tex] is separate and independent as it involves [tex]\(x^2\)[/tex] rather than [tex]\(x^3\)[/tex].
2. Combine Like Terms:
- For the [tex]\(x^3\)[/tex] terms:
[tex]\[
-4x^3 - 12x^3 = (-4 - 12)x^3 = -16x^3
\][/tex]
- The [tex]\(x^2\)[/tex] term remains as [tex]\(9x^2\)[/tex].
3. Write the Simplified Expression:
- After combining the like terms, the simplified expression is:
[tex]\[
-16x^3 + 9x^2
\][/tex]
Now, compare this simplified expression to the choices provided:
- [tex]\(x^8\)[/tex]
- [tex]\(-7x^8\)[/tex]
- [tex]\(-8x^3 + 9x^2\)[/tex]
- [tex]\(-16x^3 + 9x^2\)[/tex]
- [tex]\(-16x^6 + 9x^2\)[/tex]
The equivalent expression is [tex]\(-16x^3 + 9x^2\)[/tex], which matches the choice:
[tex]\(-16x^3 + 9x^2\)[/tex].
So, the expression equivalent to [tex]\(-4x^3 - 12x^3 + 9x^2\)[/tex] is [tex]\(\boxed{-16x^3 + 9x^2}\)[/tex].
1. Identify Like Terms:
- The terms [tex]\(-4x^3\)[/tex] and [tex]\(-12x^3\)[/tex] are like terms because they both have [tex]\(x^3\)[/tex].
- The term [tex]\(9x^2\)[/tex] is separate and independent as it involves [tex]\(x^2\)[/tex] rather than [tex]\(x^3\)[/tex].
2. Combine Like Terms:
- For the [tex]\(x^3\)[/tex] terms:
[tex]\[
-4x^3 - 12x^3 = (-4 - 12)x^3 = -16x^3
\][/tex]
- The [tex]\(x^2\)[/tex] term remains as [tex]\(9x^2\)[/tex].
3. Write the Simplified Expression:
- After combining the like terms, the simplified expression is:
[tex]\[
-16x^3 + 9x^2
\][/tex]
Now, compare this simplified expression to the choices provided:
- [tex]\(x^8\)[/tex]
- [tex]\(-7x^8\)[/tex]
- [tex]\(-8x^3 + 9x^2\)[/tex]
- [tex]\(-16x^3 + 9x^2\)[/tex]
- [tex]\(-16x^6 + 9x^2\)[/tex]
The equivalent expression is [tex]\(-16x^3 + 9x^2\)[/tex], which matches the choice:
[tex]\(-16x^3 + 9x^2\)[/tex].
So, the expression equivalent to [tex]\(-4x^3 - 12x^3 + 9x^2\)[/tex] is [tex]\(\boxed{-16x^3 + 9x^2}\)[/tex].
