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Simplify [tex]\left(3x^2 - 3 + 9x^3\right) - \left(4x^3 - 2x^2 + 16\right)[/tex].

A. [tex]x^3 - 5x^2 + 25[/tex]
B. [tex]-x^3 + x^2 - 25[/tex]
C. [tex]5x^3 + x^2 + 13[/tex]
D. [tex]5x^3 + 5x^2 - 19[/tex]

Answer :

To simplify the expression [tex]\(\left(3x^2 - 3 + 9x^3\right) - \left(4x^3 - 2x^2 + 16\right)\)[/tex], follow these steps:

1. Distribute the negative sign across the second polynomial:
Subtract each term in the second polynomial from the first:
[tex]\[
\begin{align*}
& (3x^2 - 3 + 9x^3) - (4x^3 - 2x^2 + 16) \\
= & (3x^2 - 3 + 9x^3) - 4x^3 + 2x^2 - 16.
\end{align*}
\][/tex]

2. Combine like terms:
Now, group and combine the coefficients of similar terms:

- Cubic terms ([tex]\(x^3\)[/tex]):
[tex]\[
9x^3 - 4x^3 = 5x^3
\][/tex]

- Square terms ([tex]\(x^2\)[/tex]):
[tex]\[
3x^2 + 2x^2 = 5x^2
\][/tex]

- Constant terms:
[tex]\[
-3 - 16 = -19
\][/tex]

3. Write the simplified expression:
Combine the results from each category to reach the simplified expression:
[tex]\[
5x^3 + 5x^2 - 19
\][/tex]

Thus, the simplified form of the original expression is [tex]\(5x^3 + 5x^2 - 19\)[/tex].

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