Answer :
Let's simplify each polynomial step-by-step, classify them by their degree, and the number of terms.
### Polynomial 1: [tex]\(\left(x - \frac{1}{2}\right)(6x + 2)\)[/tex]
Step-by-step expansion using the distributive property:
[tex]\[ \left( x - \frac{1}{2} \right)(6x + 2) \][/tex]
[tex]\[ = x \cdot 6x + x \cdot 2 - \frac{1}{2} \cdot 6x - \frac{1}{2} \cdot 2 \][/tex]
[tex]\[ = 6x^2 + 2x - 3x - 1 \][/tex]
[tex]\[ = 6x^2 - x - 1 \][/tex]
- Simplified Form: [tex]\(6x^2 - x - 1\)[/tex]
- Degree: 2 (quadratic)
- Number of Terms: 3 (trinomial)
### Polynomial 2: [tex]\(\left(7x^2 + 3x\right) - \frac{1}{3}\left(21x^2 - 12\right)\)[/tex]
Step-by-step expansion:
[tex]\[ \left(7x^2 + 3x\right) - \frac{1}{3}\left(21x^2 - 12\right) \][/tex]
[tex]\[ = 7x^2 + 3x - \frac{1}{3}(21x^2) + \frac{1}{3}(12) \][/tex]
[tex]\[ = 7x^2 + 3x - 7x^2 + 4 \][/tex]
[tex]\[ = 3x + 4 \][/tex]
- Simplified Form: [tex]\(3x + 4\)[/tex]
- Degree: 1 (linear)
- Number of Terms: 2 (binomial)
### Polynomial 3: [tex]\(4\left(5x^2 - 9x + 7\right) + 2\left(-10x^2 + 18x - 13\right)\)[/tex]
Step-by-step expansion:
[tex]\[ 4\left(5x^2 - 9x + 7\right) + 2\left(-10x^2 + 18x - 13\right) \][/tex]
[tex]\[ = 4 \cdot 5x^2 + 4 \cdot (-9x) + 4 \cdot 7 + 2 \cdot (-10x^2) + 2 \cdot 18x + 2 \cdot (-13) \][/tex]
[tex]\[ = 20x^2 - 36x + 28 - 20x^2 + 36x - 26 \][/tex]
[tex]\[ = 20x^2 - 20x^2 -36x + 36x + 28 - 26 \][/tex]
[tex]\[ = 2 \][/tex]
- Simplified Form: [tex]\(2\)[/tex]
- Degree: 0 (constant)
- Number of Terms: 1 (monomial)
Now let's fill out the table:
[tex]\[ \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 & \(6x^2 - x - 1\) & quadratic & trinomial \\ \hline 2 & \(3x + 4\) & linear & binomial \\ \hline 3 & \(2\) & constant & monomial \\ \hline \end{tabular} \][/tex]
So the table should look like this:
[tex]\[ \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 & \(6x^2 - x - 1\) & quadratic & trinomial \\ \hline 2 & \(3x + 4\) & linear & binomial \\ \hline 3 & \(2\) & constant & monomial \\ \hline \end{tabular} \][/tex]
### Polynomial 1: [tex]\(\left(x - \frac{1}{2}\right)(6x + 2)\)[/tex]
Step-by-step expansion using the distributive property:
[tex]\[ \left( x - \frac{1}{2} \right)(6x + 2) \][/tex]
[tex]\[ = x \cdot 6x + x \cdot 2 - \frac{1}{2} \cdot 6x - \frac{1}{2} \cdot 2 \][/tex]
[tex]\[ = 6x^2 + 2x - 3x - 1 \][/tex]
[tex]\[ = 6x^2 - x - 1 \][/tex]
- Simplified Form: [tex]\(6x^2 - x - 1\)[/tex]
- Degree: 2 (quadratic)
- Number of Terms: 3 (trinomial)
### Polynomial 2: [tex]\(\left(7x^2 + 3x\right) - \frac{1}{3}\left(21x^2 - 12\right)\)[/tex]
Step-by-step expansion:
[tex]\[ \left(7x^2 + 3x\right) - \frac{1}{3}\left(21x^2 - 12\right) \][/tex]
[tex]\[ = 7x^2 + 3x - \frac{1}{3}(21x^2) + \frac{1}{3}(12) \][/tex]
[tex]\[ = 7x^2 + 3x - 7x^2 + 4 \][/tex]
[tex]\[ = 3x + 4 \][/tex]
- Simplified Form: [tex]\(3x + 4\)[/tex]
- Degree: 1 (linear)
- Number of Terms: 2 (binomial)
### Polynomial 3: [tex]\(4\left(5x^2 - 9x + 7\right) + 2\left(-10x^2 + 18x - 13\right)\)[/tex]
Step-by-step expansion:
[tex]\[ 4\left(5x^2 - 9x + 7\right) + 2\left(-10x^2 + 18x - 13\right) \][/tex]
[tex]\[ = 4 \cdot 5x^2 + 4 \cdot (-9x) + 4 \cdot 7 + 2 \cdot (-10x^2) + 2 \cdot 18x + 2 \cdot (-13) \][/tex]
[tex]\[ = 20x^2 - 36x + 28 - 20x^2 + 36x - 26 \][/tex]
[tex]\[ = 20x^2 - 20x^2 -36x + 36x + 28 - 26 \][/tex]
[tex]\[ = 2 \][/tex]
- Simplified Form: [tex]\(2\)[/tex]
- Degree: 0 (constant)
- Number of Terms: 1 (monomial)
Now let's fill out the table:
[tex]\[ \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 & \(6x^2 - x - 1\) & quadratic & trinomial \\ \hline 2 & \(3x + 4\) & linear & binomial \\ \hline 3 & \(2\) & constant & monomial \\ \hline \end{tabular} \][/tex]
So the table should look like this:
[tex]\[ \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 & \(6x^2 - x - 1\) & quadratic & trinomial \\ \hline 2 & \(3x + 4\) & linear & binomial \\ \hline 3 & \(2\) & constant & monomial \\ \hline \end{tabular} \][/tex]