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

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.