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 by its degree and number of terms.

Polynomial 1: [tex]$\quad\left(3x - \frac{1}{4}\right)(4x + 8)$[/tex]

Polynomial 2: [tex]$\quad\left(5x^2 + 7x\right) - \frac{1}{2}\left(10x^2 - 4\right)$[/tex]

Polynomial 3: [tex]$\quad3\left(8x^2 + 4x - 2\right) + 6\left(-4x^2 - 2x + 3\right)$[/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 Polynomial 1 & & & \\
\hline Polynomial 2 & & linear & \\
\hline Polynomial 3 & 12 & & \\
\hline
\end{tabular}

monomial [tex]$\quad 3x^2 + 4x - 2 \quad 12x^2 + 23x - 2 \quad 12x + 20 \quad$[/tex] quadratic [tex]$\quad 7x + 2$[/tex]

binomial

constant



Answer :

Let's simplify each polynomial step-by-step and classify them according to their degree and number of terms.

### Polynomial 1: [tex]\((3x-\frac{1}{4})(4x+8)\)[/tex]

To simplify Polynomial 1, we'll use the distributive property (also known as FOIL for binomials):

1. First term: [tex]\( 3x \cdot 4x = 12x^2 \)[/tex]
2. Outer term: [tex]\( 3x \cdot 8 = 24x \)[/tex]
3. Inner term: [tex]\( -\frac{1}{4} \cdot 4x = -x \)[/tex]
4. Last term: [tex]\( -\frac{1}{4} \cdot 8 = -2 \)[/tex]

Combine these terms:

[tex]\[ 12x^2 + 24x - x - 2 = 12x^2 + 23x - 2 \][/tex]

This is our simplified Polynomial 1. It is a quadratic (degree 2) polynomial with three terms, also known as a trinomial.

### Polynomial 2: [tex]\(\left(5 x^2+7 x\right)-\frac{1}{2}\left(10 x^2-4\right)\)[/tex]

To simplify Polynomial 2, we first distribute and then combine like terms:

1. Distribute [tex]\(\frac{1}{2}\)[/tex] into [tex]\((10 x^2 - 4)\)[/tex]:

[tex]\[ \frac{1}{2}(10 x^2 - 4) = 5 x^2 - 2 \][/tex]

2. Subtract this result from the original polynomial:

[tex]\[ (5 x^2 + 7 x) - (5 x^2 - 2) \][/tex]

Combine the like terms:

[tex]\[ 5 x^2 + 7 x - 5 x^2 + 2 = 7 x + 2 \][/tex]

This is our simplified Polynomial 2. It is a linear (degree 1) polynomial with two terms, also known as a binomial.

### Polynomial 3: [tex]\(3(8 x^2 + 4 x - 2) + 6(-4 x^2 - 2 x + 3)\)[/tex]

To simplify Polynomial 3, distribute the constants and then combine like terms:

1. Distribute the 3 into [tex]\((8 x^2 + 4 x - 2)\)[/tex]:

[tex]\[ 3(8 x^2 + 4 x - 2) = 24 x^2 + 12 x - 6 \][/tex]

2. Distribute the 6 into [tex]\((-4 x^2 - 2 x + 3)\)[/tex]:

[tex]\[ 6(-4 x^2 - 2 x + 3) = -24 x^2 - 12 x + 18 \][/tex]

3. Combine the like terms:

[tex]\[ 24 x^2 + 12 x - 6 + (-24 x^2 - 12 x + 18) = 24x^2 - 24x^2 + 12x - 12x -6 + 18 \ = 0x^2 + 0x + 12 = 12 \][/tex]

This is our simplified Polynomial 3. It is a constant (degree 0) polynomial with one term, also known as a monomial.

### Final Classification

Let's fill in the table with the simplified forms and classifications:

\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 Polynomial 1 & [tex]\( 12 x^2 + 23 x - 2 \)[/tex] & quadratic & trinomial \\
\hline Polynomial 2 & [tex]\( 7x + 2 \)[/tex] & linear & binomial \\
\hline Polynomial 3 & 12 & constant & monomial \\
\hline
\end{tabular}