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Drag each label to the correct location on the table. Each label can be used more than once, but not all labels will be used.

Simplify the given polynomials. Then, classify each polynomial by its degree and number of terms.

[tex]\[
\begin{array}{l}
\text { Polynomial 1: } \quad\left(x-\frac{1}{2}\right)(6 x+2) \\
\text { Polynomial 2: } \quad\left(7 x^2+3 x\right)-\frac{1}{3}\left(21 x^2-12\right) \\
\text { Polynomial 3 : } \quad 4\left(5 x^2-9 x+7\right)+2\left(-10 x^2+18 x-13\right) \\
\end{array}
\][/tex]

[tex]\[
\begin{array}{l}
\text{6 } x^2+2 x-\frac{1}{2} \quad \text{ constant } \quad 2 \quad \text{ trinomial } \quad \text{ linear } \quad \text{ binomial } \quad 6 \\
\text{6 } x^2-x-1 \\
\end{array}
\][/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 & & quadratic & \\
\hline 2 & [tex]$3x + 4$[/tex] & linear & binomial \\
\hline 3 & & & monomial \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
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]
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