Answer :

To divide the polynomial [tex]\(9x^5 + 2x^4 + x^3 - 2x^2 + 2x + 23\)[/tex] by [tex]\(x + 1\)[/tex], we'll use polynomial long division. Here's a detailed step-by-step explanation:

1. Set Up the Division:
Write [tex]\(9x^5 + 2x^4 + x^3 - 2x^2 + 2x + 23\)[/tex] inside the division symbol and [tex]\(x + 1\)[/tex] outside.

2. First Term:
Divide the first term of the dividend, [tex]\(9x^5\)[/tex], by the first term of the divisor, [tex]\(x\)[/tex]. This gives [tex]\(9x^4\)[/tex].
- Multiply [tex]\(9x^4\)[/tex] with the entire divisor [tex]\(x + 1\)[/tex] to get [tex]\(9x^5 + 9x^4\)[/tex].
- Subtract [tex]\(9x^5 + 9x^4\)[/tex] from the original polynomial, giving: [tex]\(-7x^4 + x^3 - 2x^2 + 2x + 23\)[/tex].

3. Second Term:
Divide [tex]\(-7x^4\)[/tex], the new first term of the remainder, by [tex]\(x\)[/tex]. This gives [tex]\(-7x^3\)[/tex].
- Multiply [tex]\(-7x^3\)[/tex] with [tex]\(x + 1\)[/tex] to get [tex]\(-7x^4 - 7x^3\)[/tex].
- Subtract [tex]\(-7x^4 - 7x^3\)[/tex], leaving: [tex]\(8x^3 - 2x^2 + 2x + 23\)[/tex].

4. Third Term:
Divide [tex]\(8x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(8x^2\)[/tex].
- Multiply [tex]\(8x^2\)[/tex] with [tex]\(x + 1\)[/tex] to get [tex]\(8x^3 + 8x^2\)[/tex].
- Subtract, resulting in: [tex]\(-10x^2 + 2x + 23\)[/tex].

5. Fourth Term:
Divide [tex]\(-10x^2\)[/tex] by [tex]\(x\)[/tex], giving [tex]\(-10x\)[/tex].
- Multiply [tex]\(-10x\)[/tex] with [tex]\(x + 1\)[/tex] to get [tex]\(-10x^2 - 10x\)[/tex].
- Subtract to find: [tex]\(12x + 23\)[/tex].

6. Fifth Term:
Divide [tex]\(12x\)[/tex] by [tex]\(x\)[/tex], which results in [tex]\(12\)[/tex].
- Multiply [tex]\(12\)[/tex] with [tex]\(x + 1\)[/tex] to obtain [tex]\(12x + 12\)[/tex].
- Subtract, leaving the final remainder of [tex]\(11\)[/tex].

Result:
The quotient of the division is [tex]\(9x^4 - 7x^3 + 8x^2 - 10x + 12\)[/tex], and the remainder is [tex]\(11\)[/tex].

So, the expression can be written as:
[tex]\[
\frac{9x^5 + 2x^4 + x^3 - 2x^2 + 2x + 23}{x + 1} = 9x^4 - 7x^3 + 8x^2 - 10x + 12 + \frac{11}{x+1}
\][/tex]

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