Let's simplify each polynomial and classify them by their degree and number of terms.
### Polynomial 1: [tex]\(\left(x-\frac{1}{2}\right)(6x+2)\)[/tex]
#### Simplification:
[tex]\[ \left(x - \frac{1}{2}\right)(6x + 2) = x(6x + 2) - \frac{1}{2}(6x + 2) \][/tex]
[tex]\[ = 6x^2 + 2x - 3x - 1 \][/tex]
[tex]\[ = 6x^2 - x - 1 \][/tex]
So, the simplified form is [tex]\(6x^2 - x - 1\)[/tex].
#### Classification:
- Degree: quadratic (since the highest power of [tex]\(x\)[/tex] is 2)
- Number of terms: trinomial (since there are 3 terms)
### Polynomial 2: [tex]\(\left(7x^2 + 3x\right) - \frac{1}{3}\left(21x^2 - 12\right)\)[/tex]
#### Simplification:
[tex]\[ \left(7x^2 + 3x\right) - \frac{1}{3}\left(21x^2 - 12\right) = 7x^2 + 3x - \left(7x^2 - 4\right) \][/tex]
[tex]\[ = 7x^2 + 3x - 7x^2 + 4 \][/tex]
[tex]\[ = 3x + 4 \][/tex]
So, the simplified form is [tex]\(3x + 4\)[/tex].
#### Classification:
- Degree: linear (since the highest power of [tex]\(x\)[/tex] is 1)
- Number of terms: binomial (since there are 2 terms)
### Polynomial 3: [tex]\(4\left(5x^2 - 9x + 7\right) + 2\left(-10x^2 + 18x - 13\right)\)[/tex]
#### Simplification:
[tex]\[ 4\left(5x^2 - 9x + 7\right) + 2\left(-10x^2 + 18x - 13\right) = 20x^2 - 36x + 28 - 20x^2 + 36x - 26 \][/tex]
[tex]\[ = 20x^2 - 20x^2 - 36x + 36x + 28 - 26 \][/tex]
[tex]\[ = 28 - 26 \][/tex]
[tex]\[ = 2 \][/tex]
So, the simplified form is [tex]\(2\)[/tex].
#### Classification:
- Degree: constant (since there is no [tex]\(x\)[/tex] term)
- Number of terms: monomial (since there is only 1 term)
### Final Table:
\begin{tabular}{|c|c|c|c|}
\hline
Polynomial & Simplified Form & \begin{tabular}{c} Name by \\ Degree \end{tabular} & \begin{tabular}{c} Name by \\ Number of Terms \end{tabular} \\
\hline
1 & [tex]\(6x^2 - x - 1\)[/tex] & quadratic & trinomial \\
\hline
2 & [tex]\(3x + 4\)[/tex] & linear & binomial \\
\hline
3 & [tex]\(2\)[/tex] & constant & monomial \\
\hline
\end{tabular}
