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**Classifying & Simplifying Polynomials**

Simplify each polynomial and write it in standard form. Then classify it based on its degree and number of terms.

1. \[
\begin{array}{l}
18x^2 + x - 20x + 5 \\
= (x - 20x) + (18x^2 + 5) \\
= 18x^2 - 19x + 5
\end{array}
\]

2. \[
\begin{array}{l}
5x^2 - 9x^3 - 8x + x^2 \\
= (-9x^3) + (5x^2 + x^2) - 8x \\
= -9x^3 + 6x^2 - 8x
\end{array}
\]

3. \[
\begin{array}{l}
x^3 - 24 - 5x^5 + 13 \\
= (-5x^5) + x^3 + (-24 + 13) \\
= -5x^5 + x^3 - 11
\end{array}
\]

4. \[
\begin{array}{l}
-19x + 5 + 19x \\
= -19x + 19x + 5 \\
= 5
\end{array}
\]

5. \[
\begin{array}{l}
26x^4 - 9 + 3x - 17x^2 \\
= 26x^4 - 17x^2 + 3x - 9
\end{array}
\]

6. \[
\begin{array}{l}
7x - 19 - 6x - 24 + 13x^2 \\
= (7x - 6x) + 13x^2 - 43 \\
= 13x^2 + x - 43
\end{array}
\]

7. \[
\begin{array}{l}
-13x^3 - 9x + 27x^3 \\
= (-13x^3 + 27x^3) - 9x \\
= 14x^3 - 9x
\end{array}
\]

8. \[
\begin{array}{l}
4x - 18 - 5x + 17 \\
= (4x - 5x) + (-18 + 17) \\
= -x - 1
\end{array}
\]

9. \[
\begin{array}{l}
39x^3 + 18x - 1 + 5x^4 - x^2 \\
= 5x^4 + 39x^3 - x^2 + 18x - 1
\end{array}
\]

10. \[
\begin{array}{l}
-45 - \frac{1}{8}x + 30 + 10x + 15 \\
= (10x - \frac{1}{8}x) + (-45 + 30 + 15) \\
= \frac{79}{8}x + 0
\end{array}
\]

11. \[
\begin{array}{l}
11x^2 + (-x + 3x - 2x + 11x^2) \\
= 11x^2 + (11x^2 - x + 3x - 2x) \\
= 22x^2 - 13
\end{array}
\]

12. \[
\begin{array}{l}
35x^3 + 12x^8 - 22x + x^3 \\
= 12x^8 + (35x^3 + x^3) - 22x \\
= 12x^8 + 36x^3 - 22x
\end{array}
\]

Answer :

Sure! Let's simplify each polynomial, write them in standard form, and classify them based on their degree and number of terms.

### 1. [tex]\(18x^2 + x^{-1} - 20x + 5\)[/tex]
This expression seems to mistakenly include [tex]\(x^{-1}\)[/tex]. It doesn't make usual sense in polynomial context unless we're considering other special conditions (since polynomials can't have negative exponents). Let's assume it is a typo and focus on other terms:
- Simplifies to [tex]\(18x^2 - 20x + 5\)[/tex].
- Standard form: [tex]\(18x^2 - 20x + 5\)[/tex] (arrange terms from highest degree to lowest).
- Degree: 2 (highest power of [tex]\(x\)[/tex]).
- Number of terms: 3 (terms are [tex]\(18x^2\)[/tex], [tex]\(-20x\)[/tex], and [tex]\(5\)[/tex]).

### 2. [tex]\(5x^2 - 9x^3 - 8x + x^2\)[/tex]
- Combine like terms: [tex]\(5x^2 + x^2 = 6x^2\)[/tex].
- Simplify to [tex]\(-9x^3 + 6x^2 - 8x\)[/tex].
- Standard form: [tex]\(-9x^3 + 6x^2 - 8x\)[/tex].
- Degree: 3.
- Number of terms: 3.

### 3. [tex]\(x^3 - 24 - 5x^5 + 13\)[/tex]
- Combine constant terms: [tex]\(-24 + 13 = -11\)[/tex].
- Simplify to [tex]\(-5x^5 + x^3 - 11\)[/tex].
- Standard form: [tex]\(-5x^5 + x^3 - 11\)[/tex].
- Degree: 5.
- Number of terms: 3.

### 4. [tex]\(-19x + 5 + 19x\)[/tex]
- Combine like terms: [tex]\(-19x + 19x = 0\)[/tex].
- Simplify to [tex]\(5\)[/tex].
- Standard form: [tex]\(5\)[/tex].
- Degree: 0 (a constant).
- Number of terms: 1.

### 5. [tex]\(26x^4 - 9 + 3x - 17x^2\)[/tex]
- Already simplified.
- Standard form: [tex]\(26x^4 - 17x^2 + 3x - 9\)[/tex].
- Degree: 4.
- Number of terms: 4.

### 6. [tex]\(7x - 19 - 6x - 24 + 13x^2\)[/tex]
- Combine like terms: [tex]\(7x - 6x = x\)[/tex] and [tex]\(-19 - 24 = -43\)[/tex].
- Simplify to [tex]\(13x^2 + x - 43\)[/tex].
- Standard form: [tex]\(13x^2 + x - 43\)[/tex].
- Degree: 2.
- Number of terms: 3.

### 7. [tex]\(-13x^3 - 9x + 27x^3\)[/tex]
- Combine like terms: [tex]\(-13x^3 + 27x^3 = 14x^3\)[/tex].
- Simplify to [tex]\(14x^3 - 9x\)[/tex].
- Standard form: [tex]\(14x^3 - 9x\)[/tex].
- Degree: 3.
- Number of terms: 2.

### 8. [tex]\(4x - 18 - 5x + 17\)[/tex]
- Combine like terms: [tex]\(4x - 5x = -x\)[/tex] and [tex]\(-18 + 17 = -1\)[/tex].
- Simplify to [tex]\(-x - 1\)[/tex].
- Standard form: [tex]\(-x - 1\)[/tex].
- Degree: 1.
- Number of terms: 2.

### 9. [tex]\(39x^3 + 18x - 1 + 5x^4 - x^2\)[/tex]
- The expression simplifies to [tex]\(5x^4 + 39x^3 - x^2 + 18x - 1\)[/tex].
- Standard form: [tex]\(5x^4 + 39x^3 - x^2 + 18x - 1\)[/tex].
- Degree: 4.
- Number of terms: 5.

### 10. [tex]\(-45 - \frac{1}{8}x + 30 + 10x + 15\)[/tex]
- Combine constant terms: [tex]\(-45 + 30 + 15 = 0\)[/tex].
- Combine [tex]\(x\)[/tex] terms: [tex]\(10x - \frac{1}{8}x = \frac{79}{8}x\)[/tex].
- Simplify to [tex]\(\frac{79}{8}x\)[/tex].
- Standard form: [tex]\(\frac{79}{8}x\)[/tex].
- Degree: 1.
- Number of terms: 1.

### 11. [tex]\(11x^2 + (-x + 3x - 2x + 11x^2)\)[/tex]
- Combine like terms: [tex]\(-x + 3x - 2x = 0\)[/tex] and [tex]\(11x^2 + 11x^2 = 22x^2\)[/tex].
- Simplify to [tex]\(22x^2\)[/tex].
- Standard form: [tex]\(22x^2\)[/tex].
- Degree: 2.
- Number of terms: 1.

### 12. [tex]\(35x^3 + 12x^8 - 22x + x^3\)[/tex]
- Combine like terms: [tex]\(35x^3 + x^3 = 36x^3\)[/tex].
- Simplify to [tex]\(12x^8 + 36x^3 - 22x\)[/tex].
- Standard form: [tex]\(12x^8 + 36x^3 - 22x\)[/tex].
- Degree: 8.
- Number of terms: 3.

I hope this helps! Let me know if you have any questions.

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Rewritten by : Jeany

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