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What is the difference of the polynomials?

[tex]\left(5x^3 + 4x^2\right) - \left(6x^2 - 2x - 9\right)[/tex]

A. [tex]-x^3 + 6x^2 + 9[/tex]
B. [tex]-x^3 + 2x^2 - 9[/tex]
C. [tex]5x^3 - 2x^2 - 2x - 9[/tex]
D. [tex]5x^3 - 2x^2 + 2x + 9[/tex]

Answer :

Certainly! Let's find the difference between the given polynomials step-by-step:

We're given the expression:
[tex]\[
(5x^3 + 4x^2) - (6x^2 - 2x - 9)
\][/tex]

### Step 1: Distribute the Negative Sign

First, distribute the negative sign across the second polynomial. This means we'll change the sign of each term inside the parentheses:

[tex]\[
6x^2 - 2x - 9
\][/tex]

becomes:

[tex]\[
-6x^2 + 2x + 9
\][/tex]

### Step 2: Rewrite the Expression

Now we can rewrite the original problem with this step applied:

[tex]\[
5x^3 + 4x^2 - 6x^2 + 2x + 9
\][/tex]

### Step 3: Combine Like Terms

Now, combine the like terms:

- For the [tex]\(x^3\)[/tex] terms: There is only [tex]\(5x^3\)[/tex].
- For the [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 6x^2 = -2x^2\)[/tex]
- For the [tex]\(x\)[/tex] terms: There is only [tex]\(+2x\)[/tex].
- Constant terms: There is only [tex]\(+9\)[/tex].

Combine these results to give the final polynomial:

[tex]\[
5x^3 - 2x^2 + 2x + 9
\][/tex]

So, the difference of the polynomials is:

[tex]\[
5x^3 - 2x^2 + 2x + 9
\][/tex]

The answer is:

[tex]\(5x^3 - 2x^2 + 2x + 9\)[/tex]

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