Answer :

To solve the problem of dividing the polynomial [tex]\( \frac{4x^7 + 9x^6 + 3x^2}{-x^2} \)[/tex], we will perform polynomial division. Let's go through the steps:

1. Understand the Division: We're dividing each term of the polynomial [tex]\( 4x^7 + 9x^6 + 0x^5 + 0x^4 + 0x^3 + 3x^2 + 0x + 0 \)[/tex] by [tex]\(-x^2\)[/tex].

2. Divide Each Term Separately:
- Divide [tex]\(4x^7\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[
\frac{4x^7}{-x^2} = -4x^{7-2} = -4x^5
\][/tex]
- Divide [tex]\(9x^6\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[
\frac{9x^6}{-x^2} = -9x^{6-2} = -9x^4
\][/tex]
- Divide [tex]\(3x^2\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[
\frac{3x^2}{-x^2} = -3x^{2-2} = -3x^0 = -3
\][/tex]
- The terms [tex]\(0x^5\)[/tex], [tex]\(0x^4\)[/tex], [tex]\(0x^3\)[/tex], and [tex]\(0x\)[/tex] when divided by any term will remain zero.

3. Combine the Terms:
- After dividing each term, the result of the division is:
[tex]\[
-4x^5 - 9x^4 + 0x^3 + 0x^2 + 0x - 3
\][/tex]

4. Final Simplified Answer:
- The simplified polynomial is:
[tex]\[
-4x^5 - 9x^4 - 3
\][/tex]

Thus, the simplified result of the division is [tex]\(-4x^5 - 9x^4 - 3\)[/tex].

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Rewritten by : Barada