Answered

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.

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

Labels: binomial, [tex]$26 x^2+2 x-\frac{1}{2}$[/tex], 6, linear, trinomial, constant, [tex]$6 x^2-x-1$[/tex]

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
Polynomial & Simplified Form & Name by Degree & Name by Number of Terms \\
\hline
1 & & quadratic & \\
\hline
2 & $3x + 4$ & & \\
\hline
3 & & & monomial \\
\hline
\end{tabular}
\][/tex]



Answer :

GinnyAnswer
Let's simplify each polynomial step-by-step and then classify them accordingly.

### Polynomial 1:
Given:
[tex]\[ \left(x - \frac{1}{2}\right)(6x + 2) \][/tex]

First, expand the polynomial using the distributive property:
[tex]\[ (x - \frac{1}{2}) \cdot (6x + 2) = x \cdot 6x + x \cdot 2 - \frac{1}{2} \cdot 6x - \frac{1}{2} \cdot 2 \][/tex]
[tex]\[ = 6x^2 + 2x - 3x - 1 \][/tex]
[tex]\[ = 6x^2 - x - 1 \][/tex]

This is a quadratic polynomial with 3 terms.

### Polynomial 2:
Given:
[tex]\[ \left(7x^2 + 3x\right) - \frac{1}{3}\left(21x^2 - 12\right) \][/tex]

First, simplify the second term:
[tex]\[ \frac{1}{3}(21x^2 - 12) = 7x^2 - 4 \][/tex]

Now combine the terms:
[tex]\[ 7x^2 + 3x - (7x^2 - 4) \][/tex]
[tex]\[ = 7x^2 + 3x - 7x^2 + 4 \][/tex]
[tex]\[ = 3x + 4 \][/tex]

This is a linear polynomial with 2 terms.

### Polynomial 3:
Given:
[tex]\[ 4\left(5x^2 - 9x + 7\right) + 2\left(-10x^2 + 18x - 13\right) \][/tex]

Expand both terms separately:
[tex]\[ 4(5x^2 - 9x + 7) = 20x^2 - 36x + 28 \][/tex]
[tex]\[ 2(-10x^2 + 18x - 13) = -20x^2 + 36x - 26 \][/tex]

Now combine the terms:
[tex]\[ (20x^2 - 36x + 28) + (-20x^2 + 36x - 26) \][/tex]
[tex]\[ = 20x^2 - 20x^2 - 36x + 36x + 28 - 26 \][/tex]
[tex]\[ = 2 \][/tex]

This is a constant polynomial with 1 term (monomial).

### Final classification:
Putting all the information into 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]